1.R: Chapter 1 Review Exercises
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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.
- Answer
- 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.
- Answer
- False
State the domain and range of the given functions:
f=x2+2x−3, g=ln(x−5), h=1x+4
5) h
6) g
- Answer
- Domain: x>5, Range: all real numbers
7) h∘f
8) g∘f
- Answer
- Domain: x>2 and x<−4, Range: all real numbers
Find the degree, y-intercept, and zeros for the following polynomial functions.
9) f(x)=2x2+9x−5
10) f(x)=x3+2x2−2x
- Answer
- Degree of 3, y-intercept: (0,0), Zeros: 0,√3−1,−1−√3
Simplify the following trigonometric expressions.
11) tan2xsec2x+cos2x
12) cos2x−sin2x
- Answer
- cos(2x)
Solve the following trigonometric equations on the interval θ=[−2π,2π] exactly.
13) 6cos2x−3=0
14) sec2x−2secx+1=0
- Answer
- 0,±2π
Solve the following logarithmic equations.
15) 5x=16
16) log2(x+4)=3
- Answer
- 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.
17) f(x)=x2+2x+1
18) f(x)=1x
- Answer
- One-to-one; yes, the function has an inverse; inverse: f−1(x)=1x
For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.
19) f(x)=√9−x
20) f(x)=x2+3x+4
- Answer
- x≥−32,f−1(x)=−32+12√4x−7
21) 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.
22) 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.
- Answer
- a. C(x)=300+7x
b. 100 shirts
23) 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.
24) 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?
- Answer
- The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2
25) 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=ert, 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 decay rate of radiocarbon.
26) If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?
- Answer
- 78.51%
27) Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.