# 1.E: Functions and Graphs (Exercises)

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- Contributed by Gilbert Strang & Edwin “Jed” Herman
- Professor (Mathematics) at Massachusetts Institute of Technology (Strang) & University of Wisconsin-Stevens Point (Herman)
- Publisher: OpenStax CNX

## 1.1: Review of Functions

For the following exercises, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

Exercise:

1)

\(x\) | \(y) | \(x\) | \(y\) |
---|---|---|---|

-3 | 9 | 1 | 1 |

-2 | 4 | 2 | 4 |

-1 | 1 | 3 | 9 |

0 | 0 |

Solution: a. Domain = {\(−3,−2,−1,0,1,2,3\)}, range = {\(0,1,4,9\)}

b. Yes, a function

2)

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

-3 | -2 | 1 | 1 |

-2 | -8 | 2 | 8 |

-1 | -1 | 3 | -2 |

0 | 0 |

3)

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

1 | -3 | 1 | 1 |

2 | -2 | 2 | 2 |

3 | -1 | 3 | 3 |

0 | 0 |

Solution:a. Domain = {\(0,1,2,3\)}, range = {\(−3,−2,−1,0,1,2,3\)}

b. No, not a function

4)

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

1 | 1 | 5 | 1 |

2 | 1 | 6 | 1 |

3 | 1 | 7 | 1 |

4 | 1 |

5)

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

3 | 3 | 15 | 1 |

5 | 2 | 21 | 2 |

8 | 1 | 33 | 3 |

10 | 0 |

Solution: a. Domain = {\(3,5,8,10,15,21,33\)}, range = {\(0,1,2,3\)}

b. Yes, a function

6)

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

-7 | 11 | 1 | -2 |

-2 | 5 | 3 | 4 |

-2 | 1 | 6 | 11 |

0 | -1 |

Exercise:

For the following exercises, find the values for each function, if they exist, then simplify.

a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(−x)\) e. \(f(a)\) f. \(f(a+h)\)

1) \(f(x)=5x−2\)

Solution: a. \(−2\) b. \(3\) c. \(13\) d. \(−5x−2\) e. \(5a−2\) f. \(5a+5h−2\)

2) \(f(x)=4x^2−3x+1\)

3) \(f(x)=\frac{2}{x}\)

Solution: a. Undefined b. \(2\) c. \(23\) d. \(−\frac{2}{x}\) e \(\frac{2}{a}\) f. \(\frac{2}{a+h}\)

4) \(f(x)=|x−7|+8\)

5) \(f(x)=\sqrt{6x+5}\)

Solution: a. \(\sqrt{5}\) b. \(\sqrt{11}\) c. \(\sqrt{23}\) d. \(\sqrt{−6x+5}\) e. \(\sqrt{6a+5}\) f. \(\sqrt{6a+6h+5}\)

6) \(f(x)=\frac{x−2}{3x+7}\)

7) \(f(x)=9\)

Solution: a. 9 b. 9 c. 9 d. 9 e. 9 f. 9

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions.

1) \(f(x)=\frac{x}{x^2−16}\)

2) \(g(x)=\sqrt{8x−1}\)

Solution: \(x≥\frac{1}{8};y≥0;x=\frac{1}{8}\); no y-intercept

3) \(h(x)=\frac{3}{x^2+4}\)

4) \(f(x)=−1+\sqrt{x+2}\)

Solution: \(x≥−2;y≥−1;x=−1;y=−1+\sqrt{2}\)

5) \(f(x)=1x−\sqrt{9}\)

6) \(g(x)=\frac{3}{x−4}\)

Solution: \(x≠4;y≠0\); no x-intercept; \(y=−\frac{3}{4}\)

7) \(f(x)=4|x+5|\)

8) \(g(x)=\sqrt{\frac{7}{x−5}}\)\

Solution: \(x>5;y>0\); no intercepts

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=−3,−2,−1,0,1,2,3.\)

1) \(f(x)=x^2+1\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | 10 | 1 | 2 |

-2 | 5 | 2 | 5 |

-1 | 2 | 3 | 10 |

0 | 1 |

2) \(f(x)=3x−6\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | -15 | 1 | -3 |

-2 | -12 | 2 | 0 |

-1 | -9 | 3 | 3 |

0 | -6 |

Solution:

3) \(f(x)=\frac{1}{2}x+1\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | \(-\frac{1}{2}\) | 1 | \(\frac{3}{2}\) |

-2 | 0 | 2 | 2 |

-1 | \(\frac{1}{2}\) | 3 | \(\frac{5}{2}\) |

0 | 1 |

4) \(f(x)=2|x|\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | 6 | 1 | 2 |

-2 | 4 | 2 | 4 |

-1 | 2 | 3 | 6 |

0 | 0 |

Solution:

5) \(f(x)=-x^2\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | -9 | 1 | -1 |

-2 | -4 | 2 | -4 |

-1 | -1 | 3 | -9 |

0 | 0 |

6) \(f(x)=x^3\)

\(x\) |
\(y\) |
\(x\) |
\(y\) |
---|---|---|---|

-3 | -27 | 1 | 1 |

-2 | -8 | 2 | 8 |

-1 | -1 | 3 | 27 |

0 | 0 |

Solution:

Exercise:

For the following exercises, use the vertical line test to determine whether each of the given graphs represents a function. **Assume that a graph continues at both ends if it extends beyond the given grid.** If the graph represents a function, then determine the following for each graph:

Domain and range

\(x\) -intercept, if any (estimate where necessary)

\(y\)-Intercept, if any (estimate where necessary)

The intervals for which the function is increasing

The intervals for which the function is decreasing

The intervals for which the function is constant

Symmetry about any axis and/or the origin

Whether the function is even, odd, or neither

1)

2)

Solution: Function; a. Domain: all real numbers, range: \(y≥0\) b. \(x=±1\) c. \(y=1\) d. \(−1<x<0\) and \(1<x<∞ e\). \(−∞<x<−1\) and \(0<x<1\) f. Not constant g. \(y\)-axis h. Even

3)

4)

Solution: Function; a. Domain: all real numbers, range: \(−1.5≤y≤1.5\) b. \(x=0\) c. \(y=0\) d. all real numbers e. None f. Not constant g. Origin h. Odd

5)

6)

Solution: Function; a. Domain: \(−∞<x<∞\), range: \(−2≤y≤2\) b. \(x=0\) c. \(y=0\) d. \(−2<x<2\) e. Not decreasing f. \(−∞<x<−2\) and \(2<x<∞\) g. Origin h. Odd

7)

8)

Solution: Function; a. Domain: \(−4≤x≤4\), range: \(−4≤y≤4\) b. \(x=1\).2 c. \(y=4\) d. Not increasing e. \(0<x<4\) f. \(−4<x<0\) g. No Symmetry h. Neither

Exercise:

For the following exercises, for each pair of functions, find a. \(f+g\) b. \(f−g\) c. \(f⋅g\) d. \(f/g\). Determine the domain of each of these new functions.

1) \(f(x)=3x+4,g(x)=x−2\)

2) \(f(x)=x−8,g(x)=5x^2\)

Solution: a. \(5x^2+x−8\); all real numbers b. \(−5x^2+x−8\); all real numbers c. \(5x^3−40x^2\); all real numbers d. \(\frac{x−8}{5x^2}\);\(x≠0\)

3) \(f(x)=3x^2+4x+1,g(x)=x+1\)

4) \(f(x)=9−x^2,g(x)=x^2−2x−3\)

Solution: a. \(−2x+6\); all real numbers b. \(−2x^2+2x+12\); all real numbers c. \(−x^4+2x^3+12x^2−18x−27\); all real numbers d. \(−\frac{x+3}{x+1};x≠−1,3\)

5) \(f(x)=\sqrt{x},g(x)=x−2\)

6) \(f(x)=6+\frac{1}{x},g(x)=\frac{1}{x}\)

Solution: \(a. 6+\frac{2}{x};x≠0 b. 6; x≠0 c. 6x+\frac{1}{x^2};x≠0 d. 6x+1;x≠0\)

Exercise:

For the following exercises, for each pair of functions, find a. \((f∘g)(x)\) and b. \((g∘f)(x)\) Simplify the results. Find the domain of each of the results.

1) \(f(x)=3x,g(x)=x+5\)

2) \(f(x)=x+4,g(x)=4x−1\)

Solution: a. \(4x+3\); all real numbers b. \(4x+15\); all real numbers

3) \(f(x)=2x+4,g(x)=x^2−2\)

4) \(f(x)=x^2+7,g(x)=x^2−3\)

Solution:a. \(x^4−6x^2+16\); all real numbers b. \(x^4+14x^2+46\); all real numbers

5) \(f(x)=\sqrt{x}, g(x)=x+9\)

6) \(f(x)=\frac{3}{2x+1},g(x)=\frac{2}{x}\)

Solution: a. \(\frac{3x}{4+x};x≠0,−4\) b. \(\frac{4x+2}{3};x≠−12\)

7) \(f(x)=|x+1|,g(x)=x^2+x−4\)

8) The table below lists the NBA championship winners for the years 2001 to 2012.

Year | Winner |
---|---|

2001 | LA Lakers |

2002 | LA Lakers |

2003 | Sam Antonio Spurs |

2004 | Detroit Pistons |

2005 | Sam Antonio Spurs |

2006 | Miami Heat |

2007 | Sam Antonio Spurs |

2008 | Boston Celtics |

2009 | LA Lakers |

2010 | LA Lakers |

2011 | Dallas Mavericks |

2012 | Miami Heat |

- Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.
- Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.

Solution: a. Yes, because there is only one winner for each year.

b. No, because there are three teams that won more than once during the years 2001 to 2012.

9) [T] The area \(A\) of a square depends on the length of the side s.

1.Write a function \(A(s)\) for the area of a square.

2.Find and interpret \(A(6.5)\).

3.Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.

10) [T] The volume of a cube depends on the length of the sides s.

Write a function \(V(s)\) for the area of a square.

Find and interpret \(V(11.8)\).

Solution: a. \(V(s)=s^3\) b. \(V(11.8)≈1643\); a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

11) **[T]** A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost C to rent a car is a function of the hours \(t\) the car is rented plus the flat fee.

- Write the formula for the function that models this situation.
- Find the total cost to rent a car for 2 days and 7 hours.
- Determine how long the car was rented if the bill is $432.73.

12) [T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles N that can be driven depends on the amount of gas x in the tank.

1.Write a formula that models this situation.

2.Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.

3.Determine the domain and range of the function.

4.Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

Solution:

a. \(N(x)=15x\) b. i. \(N(20)=15(20)=300\); therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. \(N(15)=225\); therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: \(0≤x≤20\); range: [\(0,300\)] d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

13) [T] The volume V of a sphere depends on the length of its radius as \(V=(4/3)πr3\). Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius \(6.371×106\) m.

14) [T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by \(r(t)=6−\)[\(5/(t2+1)\)], where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

1.Express the area of the bacteria as a function of time.

2.Find the exact and approximate area of the bacterial culture in 3 hours.

3.Express the circumference of the bacteria as a function of time.

4.Find the exact and approximate circumference of the bacteria in 3 hours.

Solution: a. \(A(t)=A(r(t))=π⋅(6−\frac{5}{t^2+1})^2\) b. Exact: \(\frac{121π}{4}\); approximately 95 cm2 c. \(C(t)=C(r(t))=2π(6−\frac{5}{t^2+1})\) d. Exact: \(11π\); approximately 35 cm

15) [T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79x\), where x is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245x\), where x is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars.

1.Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?

2.Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.

16) [T] The manager at a skateboard shop pays his workers a monthly salary S of $750 plus a commission of $8.50 for each skateboard they sell.

1.Write a function \(y=S(x)\) that models a worker’s monthly salary based on the number of skateboards x he or she sells.

2.Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards.

3.Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line \(y=1400\).)

Solution: a. \(S(x)=8.5x+750\) b. $962.50, $1090, $1217.50 c. 77 skateboards

17) [T] Use a graphing calculator to graph the half-circle \(y=\sqrt{25−(x−4)^2}\). Then, use the INTERCEPT feature to find the value of both the \(x\)- and \(y\)-intercepts.

## 1.2: Basic Classes of Functions

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical.

1) \((-2,4)\) and \((1,1)\)

Solution: a. −1 b. Decreasing

2) \((-1,4)\) and \((3,-1)\)

3 \((3,5)\) and \((-1,2)\)

Solution: a. 3/4 b. Increasing

4) \((6,4)\) and \((4,-3)\)

5) \((2,3)\) and \((5,7)\)

Solution: a. 4/3 b. Decreasing

6) \((1,9)\) and \((-8,5)\)

7) \((2,4)\) and \((1,4)\)

Solution: a. 0 b. Horizontal

8) \((1,4)\) and \((1,0)\)

For the following exercises, write the equation of the line satisfying the given conditions in slope-intercept form.

1) Slope =\(−6\), passes through \((1,3)\)

Solution: \(y=−6x+9\)

2) Slope =\(3\), passes through \((-3,2)\)

3) Slope =\(\frac{1}{3}\), passes through \((0,4)\)

Solution: \(y=\frac{1}{3}x+4\)

4) Slope =\(\frac{2}{5}\), \(x\)-intercept =\(8\)

5) Passing through \((2,1\) and \((−2,−1)\)

Solution: \(y=\frac{1}{2}x\)

6) Passing through \((−3,7)\) and \((1,2)\)

7) \(x\)-intercept =\(5\) and \(y\)-intercept =\(−3\)

Solution:\(y=\frac{3}{5}x−3\)

8) \(x\)-Intercept =−\(6\) and \(y\)-intercept =\(9\)

For the following exercises, for each linear equation, a. give the slope \(m\) and \(y\)-intercept b, if any, and b. graph the line.

1) \(y=2x−3\)

Solution: a. \((m=2,b=−3)\)

b.

2) \(y=−\frac{1}{7}x+1\)

3) \(f(x)=-6x\)

a. \((m=−6,b=0)\)

b.

4) \(f(x)=−5x+4\)

5) \(4y+24=0\)

Solution: a. \((m=0,b=−6)\)

b.

6) \(8x-4=0\)

7) \(2x+3y=6\)

Solution: a. \((m=−\frac{2}{3},b=2)\)

b.

8) \(6x−5y+15=0\)

For the following exercises, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the \(y\)-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.

1) \(f(x)=2x^2−3x−5\)

Solution: a. 2 b. \(\frac{5}{2}\),−1; c. −5 d. Both ends rise e. Neither

2) \(f(x)=−3x^2+6x\)

3) \(f(x)=\frac{1}{2}x^2−1\)

Solution: a. 2 b. ±\(\sqrt{2}\) c. −1 d. Both ends rise e. Even

4) \(f(x)=x^3+3x^2−x−3\)

5) \(f(x)=3x−x^3\)

Solution: a. 3 b. 0, ±\(\sqrt{3}\) c. 0 d. Left end rises, right end falls e. Odd

For the following exercises, use the graph of \(f(x)=x^2\) to graph each transformed function \(g\).

Exercise:

1) \(g(x)=x^2−1\)

2) \(g(x)=(x+3)^2+1\)

Solution:

For the following exercises, use the graph of \(f(x)=\sqrt{x}\) to graph each transformed function \(g\).

1) \(g(x)=\sqrt{x+2}\)

2) \(g(x)=−\sqrt{x}−1\)

For the following exercises, use the graph of \(y=f(x)\) to graph each transformed function \(g\).

1) \(g(x)=f(x)+1\)

2) \(g(x)=f(x−1)+2\)

Solution:

For the following exercises, for each of the piecewise-defined functions, a. evaluate at the given values of the independent variable and b. sketch the graph.

1) \(f(x)=\begin{cases}4x+3, & x≤0\\ -x+1, & x>0\end{cases} ;f(−3);f(0);f(2)\)

2) \(f(x)=\begin{cases}x^2-3, & x≤0\\ 4x+3, & x>0\end{cases} ;f(−4);f(0);f(2)\)

Solution: a. \(13,−3,5\)

b.

3) \(h(x)=\begin{cases}x+1, &x≤5\\4, &x>5\end{cases} ;h(0);h(π);h(5)\)

4) \(g(x)=\begin{cases}\frac{3}{x−2}, &x≠2\\4, &x=2\end{cases} ;g(0);g(−4);g(2)\)

Solution: a. \(\frac{−3}{2},\frac{−1}{2},4\)

b.

For the following exercises, determine whether the statement is *true or false*. Explain why.

1) \(f(x)=(4x+1)/(7x−2)\) is a transcendental function.

2) \(g(x)=\sqrt[3]{x}\) is an odd root function

Solution: True; \(n=3\)

3) A logarithmic function is an algebraic function.

4) A function of the form \(f(x)=x^b\), where \(b\) is a real valued constant, is an exponential function.

Solution: False; \(f(x)=x^b\), where \(b\) is a real-valued constant, is a power function

5) The domain of an even root function is all real numbers.

6) [T] A company purchases some computer equipment for $20,500. At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300.

1.Find a function \(y=V(t)\) that determines the value V of the equipment at the end of t years.

2.Find and interpret the meaning of the \(x\)- and \(y\)-intercepts for this situation.

3.What is the value of the equipment at the end of 5 years?

4.When will the value of the equipment be $3000?

Solution: a. \(V(t)=−2733t+20500\) b. \((0,20,500)\) means that the initial purchase price of the equipment is $20,500; \((7.5,0)\) means that in 7.5 years the computer equipment has no value. c. $6835 d. In approximately 6.4 years

7) [T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 \((t=0)\),total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion.

1. Find a linear function S that estimates the total online holiday sales in the year t.

2. Interpret the slope of the graph of S.

3. Use part a. to predict the year when online shopping during Christmas will reach $60 billion.

8) [T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $0.75 to make each cupcake. The owner is interested in determining the total cost \(C\) as a function of number of cupcakes made.

1.Find a linear function that relates cost C to x, the number of cupcakes made.

2.Find the cost to bake 160 cupcakes.

3.If the owner sells the cupcakes for $1.50 apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)

Solution: a. \(C=0.75x+125\) b. $245 c. 167 cupcakes

9) [T] A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years.

1. Find a linear function that models the price P of the house versus the number of years t since the original purchase.

2. Interpret the slope of the graph of P.

3. Find the price of the house 15 years from when it was originally purchased.

10) [T] A car was purchased for $26,000. The value of the car depreciates by $1500 per year.

1. Find a linear function that models the value V of the car after t years.

2. Find and interpret \(V(4)\).

Solution: a. \(V(t)=−1500t+26,000\) b. In 4 years, the value of the car is $20,000.

11) [T] A condominium in an upscale part of the city was purchased for $432,000. In 35 years it is worth $60,500. Find the rate of depreciation.

12) [T] The total cost C (in thousands of dollars) to produce a certain item is modeled by the function \(C(x)=10.50x+28,500\), where x is the number of items produced. Determine the cost to produce 175 items.

Solution: $30,337.50

13) [T] A professor asks her class to report the amount of time t they spent writing two assignments. Most students report that it takes them about 45 minutes to type a four-page assignment and about 1.5 hours to type a nine-page assignment.

1. Find the linear function \(y=N(t)\) that models this situation, where \(N\) is the number of pages typed and t is the time in minutes.

2. Use part a. to determine how many pages can be typed in 2 hours.

3. Use part a. to determine how long it takes to type a 20-page assignment.

14) [T] The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function \(P(t)=1.8576t+68.052\), where t is time in years and \(t=0\) corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.

Solution: 96% of the total capacity

15) [T] The admissions office at a public university estimates that 65% of the students offered admission to the class of 2019 will actually enroll.

1. Find the linear function \(y=N(x)\), where \(N\) is the number of students that actually enroll and \(x\) is the number of all students offered admission to the class of 2019.

2. If the university wants the 2019 freshman class size to be 1350, determine how many students should be admitted.

## 1.3: Trigonometric Functions

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of \(π\).

1) \(240°\)

Solution: \(\frac{4π}{3} rad\)

2) \(15°\)

3) \(60°\)

Solution: \(\frac{-π}{3} rad\)

4) \(-225°\)

5) \(330°\)

Solution: \(\frac{11π}{6} rad\)

For the following exercises, convert each angle in radians to degrees.

1) \(\frac{π}{2} rad\)

2) \(\frac{7π}{6} rad\)

Solution: \(210°\)

3) \(\frac{11π}{2} rad\)

4) \(-3π rad\)

Solution:\(-540°\)

5) \(\frac{5π}{12} rad\)

Evaluate the following functional values.

1) \(cos(\frac{4π}{3}\))

Solution: -0.5

2) \(tan(\frac{19π}{4}\))

3) \(sin(-\frac{3π}{4}\))

Solution: \(-\frac{sqrt{2}}{2}\)

4) \(sec(-\frac{π}{6}\))

5) \(sin(-\frac{π}{12}\))

Solution: \(\frac{\sqrt{3}-1}{2\sqrt{2}}\)

6) \(cos(-\frac{5π}{12}\))

For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place.

225°=225°⋅π180°=5π4

1) \(a=4, c=7)\)

Solution: \(a. b=5.7 b. sinA=\frac{4}{7},cosA=\frac{5.7}{7},tanA=\frac{4}{5.7} ,cscA=\frac{7}{4} ,secA=\frac{7}{5.7} ,cotA=\frac{5.7}{4}\)

2) \(a=21, c=29)\)

3) \(a=85.3, b=125.5)\)

Solution: \(a. c=151.7 b. sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471\)

4) \(b=40, c=41)\)

5) \(a=84, b=13)\)

Solution: \(a. c=85 b. sinA=\frac{84}{85},cosA=\frac{13}{85}, tanA=\frac{84}{13}, cscA=\frac{85}{84} ,secA=\frac{85}{13} ,cotA=\frac{13}{84}\)

6) \(b=28, c=35)\)

For the following exercises, \(P\) is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle \(θ\) with a terminal side that passes through point \(P\). Rationalize denominators.

1) \(P(\frac{7}{25},y), y>0\)

Solution:\(a.y=\frac{24}{25}b.sinθ=\frac{24}{25},cosθ=\frac{7}{25},tanθ=\frac{24}{7},cscθ=\frac{25}{24} , secθ=\frac{25}{7},cotθ=\frac{7}{24}\)

2) \(P(\frac{-15}{17},y), y>0\)

3) \(P(\frac{x}{\frac{\sqrt{7}}{3}}), y>0\)

Solution: a. \(x=−\frac{\sqrt{2}}{3} b. sinθ=\frac{\sqrt{7}}{3} ,cosθ=\frac{−\sqrt{2}}{3} ,tanθ=\frac{\sqrt{−14}}{2},cscθ=\frac{3\sqrt{7}}{7},secθ=\frac{−3\sqrt{2}}{2} ,cotθ=\frac{−\sqrt{14}}{7}\)

4) \(P(\frac{x}{\frac{-\sqrt{15}}{4}}), y>0\)

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

1) \(tan^2x+sinxcscx\)

Solution: \(sec^2x\)

2) \(secxsinxcotx\)

3)\(\frac{tan^2x}{sec^2x}\)

Solution: \(sin^2x\)

4) \(secx-cosx\)

5) \((1+tanθ)^2-2tanθ\)

Solution: sex^2θ

6) \(sinx(cscx-sinx)\)

7) \(\frac{cos t}{sin t}+\frac{sin t}{1+cos t}\)

Solution: \(1/sin t) = csc t\)

8) \(\frac{1+tan^2α}{1+cot^2α}\)

For the following exercises, verify that each equation is an identity.

1) \(\frac{tanθcotθ}{cscθ}=sinθ\)

2) \(\frac{sec^2θ}{tanθ}=secθcscθ\)

3) \(\frac{sin t}{csc t} + \frac{cos t}{sec t} = 1\)

4) \(\frac{sinx}{cosx+1}+\frac{cosx−1}{sinx}=0\)

5) \(cotγ+tanγ=secγcscγ\)

6) \(sin^2β+tan^2β+cos^2β=sec^2β\)

7) \(\frac{1}{1−sinα}+\frac{1}{1+sinα}=2sec^2α\)

8)\(\frac{tanθ−cotθ}{sinθcosθ}=sec^2θ−csc^2θ\)

For the following exercises, solve the trigonometric equations on the interval \(0≤θ<2π.\)

1) \(2sinθ−1=0\)

Solution: {\(\frac{π}{6},\frac{5π}{6}\)}

2) \(1+cosθ=\frac{1}{2}\)

3) \(2tan^2θ=2\)

Solution: {\(\frac{π}{4},\frac{3π}{4},\frac{5π}{4},\frac{7π}{4}\)}

4) \(4sin^2θ−2=0\)

5) \(\sqrt{3}cotθ+1=0\)

Solution: {\(\frac{2π}{3},\frac{5π}{3}\)}

6) \(3secθ−2\sqrt{3}=0\)

7) \(2cosθsinθ=sinθ\)

Solution: {\(0,π,\frac{π}{3},\frac{5π}{3}\)}

8) \(csc^2θ+2cscθ+1=0\)

For the following exercises, each graph is of the form \(y=AsinBx\) or \(y=AcosBx\), where \(B>0\). Write the equation of the graph.

1)

Solution: \(y=4sin(\frac{π}{4}x)\)

2)

3)

Solution: \(y=cos(2πx)\)

4)

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

1) \(y=sin(x−\frac{π}{4})\)

Solution: \(a. 1 b. 2π c. \frac{π}{4}\) units to the right

2) \(y=3cos(2x+3)\)

3) \(y=−\frac{1}{2}sin(\frac{1}{4}x)\)

Solution: \(a. \frac{1}{2} b. 8π c. No phase shift\)

4) \(y=2cos(x−\frac{π}{3})\)

5) \(y=−3sin(πx+2)\)

Solution: \( a. 3 b. 2 c. \frac{2}{π}\) units to the left

6) \(y=4cos(2x−\frac{π}{2})\)

#### Exercise

1) [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of \(120\)°, how many inches does it move? Approximate to the nearest whole inch.

Solution: Approximately 42 in.

2) [T] Find the length of the arc intercepted by central angle \(θ\) in a circle of radius \(r\). Round to the nearest hundredth.

a. \(r=12.8\) cm, \(θ=5π6\) rad b. \(r=4.378\) cm, \(θ=7π6\) rad c. \(r=0.964\) cm, \(θ=50\)° d. \(r=8.55\) cm, \(θ=325\)°

3) [T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, \(ω\), and is given by \(ω=θ/t\), where \(θ\) is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. \(θ=\frac{7π}{4}\) rad, \(t=10\) sec b. \(θ=\frac{3π}{5}\) rad, \(t=8\) sec c. \(θ=\frac{2π}{9}\) rad, \(t=1\) min d. \(θ=23.76\) rad, \(t=14\) min

Solution: \(a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min\)

4) [T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

a)Find the radius of the circular land area.

b)If the land area is to form a \(45\)° sector of a circle instead of a whole circle, find the length of the curved side.

5) [T] The area of an isosceles triangle with equal sides of length x is \(\frac{1}{2}x^2sinθ\),

where \(θ\) is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle \(θ=5π/12\) rad.

Solution: \(≈30.9in^2\)

6) [T] A particle travels in a circular path at a constant angular speed \(ω\). The angular speed is modeled by the function \(ω=9|cos(πt−π/12)|\). Determine the angular speed at \(t=9\) sec.

7) [T] An alternating current for outlets in a home has voltage given by the function

\(V(t)=150cos368t\),

where V is the voltage in volts at time t in seconds.

a) Find the period of the function and interpret its meaning.

b) Determine the number of periods that occur when 1 sec has passed.

Solution: a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

8) [T] The number of hours of daylight in a northeast city is modeled by the function

\(N(t)=12+3sin[\frac{2π}{365}(t−79)]\),

where t is the number of days after January 1.

a) Find the amplitude and period.

b) Determine the number of hours of daylight on the longest day of the year.

c) Determine the number of hours of daylight on the shortest day of the year.

d) Determine the number of hours of daylight 90 days after January 1.

e) Sketch the graph of the function for one period starting on January 1.

9) [T] Suppose that \(T=50+10sin[\frac{π}{12}(t−8)]\) is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.

a) Determine the amplitude and period.

b) Find the temperature 7 hours after midnight.

c) At what time does \(T=60\)°?

d) Sketch the graph of \(T\) over \(0≤t≤24\).

Solution: a. Amplitude = \(10;period=24\) b. \(47.4°F\) c. 14 hours later, or 2 p.m. d.

10) [T] The function \(H(t)=8sin(\frac{π}{6}t)\) models the height H (in feet) of the tide t hours after midnight. Assume that \(t=0\) is midnight.

a) Find the amplitude and period.

b) Graph the function over one period.

c) What is the height of the tide at 4:30 a.m.?

## 1.4: Inverse Functions

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

1)

Solution: Not one-to-one

2)

3)

Solution: Not one-to-one

4)

5)

Solution: One-to-one

6)

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

1) \(f(x)=x^2−4 ,x≥0\)

Solution: a. \(f^{−1}(x)=\sqrt{x+4}\) b. Domain : \(x≥−4\),range:\(y≥0\)

2) \(f(x)=\sqrt[3]{x−4}\)

3) \(f(x)=^3+1\)

Solution: a. \(f^{−1}(x)=\frac{3}{x−1}\) b. Domain: all real numbers, range: all real numbers

4) \(f(x)=(x−1)^2, x≤1\)

5) \(f(x)=\sqrt{x−1}\)

Solution: a. \(f^{−1}(x)=x^2+1\), b. Domain: \(x≥0\), range: \(y≥1\)

6) \(f(x)=\frac{1}{x+2}\)

For the following exercises, use the graph of f to sketch the graph of its inverse function.

1)

Solution

2)

3)

Solution:

4)

For the following exercises, use composition to determine which pairs of functions are inverses.

1) \(f(x)=8x, g(x)=\frac{x}{8}\)

Solution: These are inverses.

2) \(f(x)=8x+3, g(x)=\frac{x-3}{8}\)

3) \(f(x)=5x−7,g(x)=\frac{x+5}{7}\)

Solution: These are not inverses.

4) \(f(x)=\frac{2}{3}x+2, g(x)=\frac{3}{2}x+3\)

5) \(f(x)=\frac{1}{x−1}, x≠1, g(x)=\frac{1}{x}+1,x≠0\)

Solution: These are inverses.

6) \(f(x)=x^3+1,g(x)=(x−1)^{1/3}\)

7) \(f(x)=x^2+2x+1,x≥−1, g(x)=−1+\sqrt{x},x≥0\)

Solution: These are inverses.

8) \(f(x)=\sqrt{4−x^2},0≤x≤2, g(x)=\sqrt{4−x^2},0≤x≤2\)

For the following exercises, evaluate the functions. Give the exact value.

1) \(tan^{−1}(\frac{\sqrt{3}}{3})\)

Solution: \(\frac{π}{6}\)

2) \(cos^{−1}(−\frac{\sqrt{2}}{2})\)

3) \(cot^{−1}(1)\)

Solution: \(\frac{π}{4}\)

4) \(sin^{−1}(−1)\)

5) \(cos^{−1}(\frac{\sqrt{3}}{2})\)

Solution: \(\frac{π}{6}\)

6) \(cos(tan^{−1}(\sqrt{3}))\)

7) \(sin(cos^{−1}(\frac{\sqrt{2}}{2}))\)

Solution: \(\frac{\sqrt{2}}{2}\)

8) \(sin^{−1}(sin(\frac{π}{3}))\)

9) \(tan^{−1}(tan(−\frac{π}{6}))\)

Solution: \(-\frac{π}{6}\)

#### Exercise:

1) The function \(C=T(F)=(5/9)(F−32)\) converts degrees Fahrenheit to degrees Celsius.

a) Find the inverse function \(F=T^{−1}(C)\)

b) What is the inverse function used for?

2) [T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function \(V=f(x)=500(0.04−x^2)\) for \(0≤x≤0.2.\)

a) Find \(x=f^{−1}(V).\)

b) Interpret what the inverse function is used for.

c) Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.

Solution: a. \(x=f^{−1}(V)\)=\sqrt{0.04−\frac{V}{500}}\) b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. c. 0.1 cm; 0.14 cm; 0.17 cm

3) A function that converts dress sizes in the United States to those in Europe is given by \(D(x)=2x+24.\)

a) Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States.

b) Find the function that converts European dress sizes to U.S. dress sizes.

c) Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.

4) [T] The cost to remove a toxin from a lake is modeled by the function \(C(p)=75p/(85−p),\) where \(C\) is the cost (in thousands of dollars) and \(p\) is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.

a) Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.

b) Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000.

Solution: a. $31,250, $66,667, $107,143 b. (\(p=\frac{85C}{C+75}\)) c. 34 ppb

5) [T] A race car is accelerating at a velocity given by \(v(t)=\frac{25}{4}t+54,\)

where v is the velocity (in feet per second) at time t.

a) Find the velocity of the car at 10 sec.

b) Find the inverse function.

c) Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.

6) [T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by \(μ=2sin^{−1}(\frac{1}{M}).\)

Find the Mach angle (to the nearest degree) for the following Mach numbers.

a. μ=1.4

b. μ=2.8

c. μ=4.3

Solution: a. \(~92°\) b. \(~42°\) c. \(~27°\)

7) [T] Using \(μ=2sin^{−1}(\frac{1}{M})\), find the Mach number M for the following angles.

a. μ=\(\frac{π}{6}\)

b. μ=\(\frac{2π}{7}\)

c. μ=\(\frac{3π}{8}\)

8) [T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

\(T(x)=5+18sin[\frac{π}{}6(x−4.6)],\)

where \(x\) is time in months and \(x=1.00\) corresponds to January 1. Determine the month and day when the temperature is \(21°C.\)

Solution: \(x≈6.69,8.51\); so, the temperature occurs on June 21 and August 15

9) [T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function \(D(t)=5sin(\frac{π}{6}t−\frac{7π}{6})+8,\) where \(t\) is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.

10) [T] An object moving in simple harmonic motion is modeled by the function \(s(t)=−6cos(\frac{πt}{2}),\) where \(s\) is measured in inches and t is measured in seconds. Determine the first time when the distance moved is 4.5 ft.

Solution: \(~1.5\) sec

11) [T] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle \(θ\) can be modeled by the function \(θ=tan^{−1}\frac{5.5}{x}−tan^{−1}\frac{2.5}{x}\), where \(x\) is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.

12) [T] Use a calculator to evaluate \(tan^{−1}(tan(2.1))\) and \(cos^{−1}(cos(2.1))\). Explain the results of each.

Solution: \(tan^{−1}(tan(2.1))≈−1.0416\); the expression does not equal \(2.1\) since \(2.1>1.57=\frac{π}{2}\)—in other words, it is not in the restricted domain of \(tanx\). \(\cos^{−1}(cos(2.1))=2.1\), since \(2.1\) is in the restricted domain of \(cosx\).

13) [T] Use a calculator to evaluate \(sin(sin^{−1}(−2))\) and \(tan(tan^{−1}(−2))\). Explain the results of each.

## 1.5: Exponential and Logarithmic Functions

For the following exercises, evaluate the given exponential functions as indicated, accurate to two significant digits after the decimal.

1) \(f(x)=5^x\) a. \(x=3\) b. \(x=\frac{1}{2}\) c. \(x=\sqrt{2}\)

Solution: \(a. 125 b. 2.24 c. 9.74\)

2) \(f(x)=(0.3)^x\) a. \(x=−1\) b. \(x=4\) c. \(x=−1.5\)

3) \(f(x)=10^x\) a. \(x=−2\) b. \(x=4\) c. \(x=\frac{5}{3}\)

Solution: \(a. 0.01 b. 10,000 c. 46.42\)

4) \(f(x)=e^x\) a. \(x=2\) b. \(x=−3.2\) c. \(x=π\)

For the following exercises, match the exponential equation to the correct graph.

a. \(y=4^{−x}\)

b. \(y=3^{x−1}\)

c. \(y=2^{x+1}\)

d. \(y=(\frac{1}{2})^x+2\)

e. \(y=−3^{−x}\)

f. \(y=1−5^x\)

1)

Solution: d

2)

3)

Solution: b

4)

5)

Solution: e

6)

For the following exercises, sketch the graph of the exponential function. Determine the domain, range, and horizontal asymptote.

1) \(f(x)=e^x+2\)

Solution: Domain: all real numbers, range: \((2,∞),y=2\)

2) \(f(x)=−2^x\)

3) \(f(x)=3^{x+1}\)

Solution: Domain: all real numbers, range: \((0,∞),y=0\)

4) \(f(x)=4^x−1\)

5) \(f(x)=1−2^{−x}\)

Solution: Domain: all real numbers, range: \((−∞,1),y=1\)

6) \(f(x)=5^{x+1}+2\)

7) \(f(x)=e^{−x}−1\)

Solution: Domain: all real numbers, range: \((−1,∞),y=−1\)

For the following exercises, write the equation in equivalent exponential form.

1) \(log_381=4\)

2) \(log_82=\frac{1}{3}\)

Solution: \(8^{1/3}=2\)

3) \(log_51=0\)

4) \(log_525=2\)

Solution: \(5^2=25\)

5) \(log0.1=−1\)

6) \(ln(\frac{1}{e^3})=−3\)

Solution: \(e^{−3}=\frac{1}{e^3}\)

7) \(log_93=0.5\)

8) \(ln1=0\)

Solution: \(e^0=1\)

For the following exercises, write the equation in equivalent logarithmic form.

1) \(2^3=8\)

2) \(4^{−2}=\frac{1}{16}\)

Solution: \(log_4(\frac{1}{16})=−2\)

3) \(10^2=100\)

4) \(9^0=1\)

Solution: \(log_91=0\)

5) \((\frac{1}{3})^3=\frac{1}{27}\)

6) \(\sqrt[3]{64}=4\)

Solution: \(log_{64}4=\frac{1}{3}\)

7) \(e^x=y\)

8) \(9^y=150\)

Solution: \(log_9150=y\)

9) \(b^3=45\)

10) \(4^{-3/2}=0.125\)

Solution: \(log_40.125=−\frac{3}{2}\)

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote.

1) \(f(x)=3+lnx\)

2) \(f(x)=ln(x−1)\)

Solution: Domain: \((1,∞)\), range: \((−∞,∞),x=1\)

3) \(f(x)=ln(−x)\)

4) \(f(x)=1−lnx\)

Solution: Domain: \((0,∞)\), range: \((−∞,∞),x=0\)

5) \(f(x)=\log x−1\)

6) \(f(x)=ln(x+1)\)

Solution: Domain: \((−1,∞)\), range: \((−∞,∞)\), \(x=−1\)

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms.

1) \(logx^4y\)

2) \(log_3\frac{9a^3}{b}\)

Solution: \(2+3log_3a−log_3b\)

3) \(lna\sqrt[3]{b}\)

4) \(log_5\sqrt{125xy^3}\)

Solution: \(\frac{3}{2}+\frac{1}{2}log_5x+\frac{3}{2}log_5y\)

5) \(log_\frac{\sqrt[3]{xy}}{64}\)

6) \(ln(\frac{6}{\sqrt{e^3}})\)

Solution: \(−\frac{3}{2}+ln6\)

For the following exercises, solve the exponential equation exactly.

1) \(5^x=125\)

2) \(e^{3x}−15=0\)

Solution: \(\frac{ln15}{3}\)

3) \(8^x=4\)

4) \(4^{x+1}−32=0\)

Solution: \(\frac{3}{2}\)

5) \(3^{x/14}=\frac{1}{10}\)

6) \(10^x=7.21\)

Solution: \(log7.21\)

7) \(4⋅2^{3x}−20=0\)

8) \(7^{3x−2}=11\)

Solution: \(\frac{2}{3}+\frac{log11}{3log7}\)

For the following exercises, solve the logarithmic equation exactly, if possible.

1) \(log_3x=0\)

2) \(log_5x=−2\)

Solution: \(x=\frac{1}{25}\)

3) \(log_4(x+5)=0\)

4) \(log(2x−7)=0\)

Solution: \(x=4\)

5) \(ln\sqrt{x+3}=2\)

6) \(log_6(x+9)+log_6x=2\)

Solution: \(x=3\)

7) \(log_4(x+2)−log_4(x−1)=0\)

8) \(lnx+ln(x−2)=ln4\)

Solution: \(1+\sqrt{5}\)

For the following exercises, use the change-of-base formula and either base 10 or base e to evaluate the given expressions. Answer in exact form and in approximate form, rounding to four decimal places.

1) \(log_547\)

2) \(log_782\)

Solution: \((\frac{log82}{log7}≈2.2646)\)

3) \(log_6103\)

4) \(log_{0.5}211\)

Solution: \((\frac{log211}{log0.5}≈−7.7211)\)

5) \(log_2π\)

6) \(log_{0.2}0.452\)

Solution: \((\frac{log0.452}{log0.2}≈0.4934)\)

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1) Rewrite the following expressions in terms of exponentials and simplify.

a. \(2cosh(lnx)\) b. \(cosh4x+sinh4x\) c. \(cosh2x−sinh2x\) d. \(ln(coshx+sinhx)+ln(coshx−sinhx)\)

2) [T] The number of bacteria N in a culture after t days can be modeled by the function \(N(t)=1300⋅(2)^{t/4}\). Find the number of bacteria present after 15 days.

Solution: \(~17,491\)

3) [T] The demand D (in millions of barrels) for oil in an oil-rich country is given by the function \(D(p)=150⋅(2.7)^{−0.25p}\), where p is the price (in dollars) of a barrel of oil. Find the amount of oil demanded (to the nearest million barrels) when the price is between $15 and $20.

4) [T] The amount A of a $100,000 investment paying continuously and compounded for t years is given by \(A(t)=100,000⋅e^{0.055t}\). Find the amount A accumulated in 5 years.

Solution: Approximately $131,653 is accumulated in 5 years.

5) [T] An investment is compounded monthly, quarterly, or yearly and is given by the function \(A=P(1+\frac{j}{n})^{nt}\), where \(A\) is the value of the investment at time \(t\), \(P\) is the initial principle that was invested, \(j\) is the annual interest rate, and n is the number of time the interest is compounded per year. Given a yearly interest rate of 3.5% and an initial principle of $100,000, find the amount \(A\) accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.

6) [T] The concentration of hydrogen ions in a substance is denoted by \([H+]\), measured in moles per liter. The pH of a substance is defined by the logarithmic function \(pH=−log[H+]\). This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.

a. Find the pH of the following substances. Round answers to one digit.

b. Determine whether the substance is an acid or a base.

i. Eggs: \([H+]=1.6×10^{−8}\) mol/L

ii. Beer: \([H+]=3.16×10^{−3}\) mol/L

iii. Tomato Juice: \([H+]=7.94×10^{−5}\) mol/L

Solution: i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

7) [T] Iodine-131 is a radioactive substance that decays according to the function \(Q(t)=Q_0⋅e^{−0.08664t}\), where \(Q_0\) is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.

8) [T] According to the World Bank, at the end of 2013 \((t=0)\) the U.S. population was 316 million and was increasing according to the following model:

\(P(t)=316e^{0.0074t}\),

where P is measured in millions of people and t is measured in years after 2013.

a. Based on this model, what will be the population of the United States in 2020?

b. Determine when the U.S. population will be twice what it is in 2013.

Solution: a. \(~333\) million b. 94 years from 2013, or in 2107

9) [T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function \(A(t)=1000(1.04)^t\).

a. Find the amount accumulated after 5 years and 10 years.

b. Determine how long it takes for the original investment to triple.

10) [T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.

a. Use the exponential function \(Q=Q_0e^{kt}\)to determine the value \(k\), which is the growth rate of the bacteria. Round to four decimal places.

b. Determine approximately how long it takes for 200,000 bacteria to grow.

Solution: a. \(k≈0.0578\) b. ≈\(92\) hours

11) [T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.

a. Use the exponential function \(P=P_0a^t\) to determine the growth rate constant \(a\). Round to four decimal places.

b. Use the function in part a. to determine approximately how long it takes for the rabbit population to reach 3500.

12) [T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?

Solution: The San Francisco earthquake had \(10^{3.4} or ~2512\) times more energy than the Japan earthquake.

## Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

1) A function is always one-to-one.

2) \(f∘g=g∘f\), assuming f and g are functions.

Solution: False

3) A relation that passes the horizontal and vertical line tests is a one-to-one function.

4) A relation passing the horizontal line test is a function.

Solution: False

For the following problems, state the domain and range of the given functions:

\(f=x^2+2x−3\), \(g=ln(x−5)\), \(h=\frac{1}{x+4}\)

1) h

2) g

Solution: Domain: \(x>5\), range: all real numbers

3) \(h∘f\)

4) \(g∘f\)

Solution: Domain: \(x>2\) and \(x<−4\), range: all real numbers

Find the degree, y-intercept, and zeros for the following polynomial functions.

1) \(f(x)=2x^2+9x−5\)

2) \(f(x)=x^3+2x^2−2x\)

Solution: Degree of 3, \(y\)-intercept: 0, zeros: \(0, \sqrt{3}−1,−1−\sqrt{3}\)

Simplify the following trigonometric expressions.

1) \(\frac{tan^2x}{sec^2x}+{cos^2x}\)

2) \(cos(2x)=sin^2x\)

Solution: \(cos(2x)\) or \(\frac{1}{2}(cos(2x)+1)\)

Solve the following trigonometric equations on the interval \(θ=[−2π,2π]\) exactly.

1) \(6cos2x−3=0\)

2) \(sec^2x−2secx+1=0\)

Solution: \(0,±2π\)

Solve the following logarithmic equations.

1) \(5^x=16\)

2) \log_2(x+4)=3\)

Solution: 4

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse \(f^{−1}(x)\) of the function. Justify your answer.

1) \(f(x)=x^2+2x+1\)

2) \(f(x)=\frac{1}{x}\)

Solution: One-to-one; yes, the function has an inverse; inverse: \(f^{−1}(x)=\frac{1}{y}\)

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

1) \(f(x)=\sqrt{9−x}\)

2) \(f(x)=x^2+3x+4\)

Solution: \(x≥−\frac{3}{2},f^{−1}(x)=−\frac{3}{2}+\frac{1}{2}\sqrt{4y−7}\)

3) A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and $1000 for 100 shirts.

1) a. Find the equation \(C=f(x)\) that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

Solution: a. \(C(x)=300+7x\) b. 100 shirts

2) a. Find the inverse function \(x=f^{−1}(C)\) and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend.

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

1) The population can be modeled by \(P(t)=82.5−67.5cos[(π/6)t]\), where \(t\) is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

Solution: The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

2) In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as \(P(t)=82.5−67.5cos[(π/6)t]+t\), where t is time in months (\(t=0\) represents January 1) and \(P\) is population (in thousands). When is the first time the population reaches 200,000?

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{rt}\), where \(y\) is the percentage of radiocarbon still present in the material, t is the number of years passed, and \(r=−0.0001210\) is the d78.51%ecay rate of radiocarbon.

1) If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

Solution: 78.51%

2) Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?