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Mathematics LibreTexts

1.R: Chapter 1 Review Exercises

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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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) fg=gf, 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+2x3, g=ln(x5), h=1x+4

5) h

6) g

Answer
Domain: x>5, Range: all real numbers

7) hf

8) gf

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+9x5

10) f(x)=x3+2x22x

Answer
Degree of 3, y-intercept: (0,0), Zeros: 0,31,13

Simplify the following trigonometric expressions.

11) tan2xsec2x+cos2x

12) cos2xsin2x

Answer
cos(2x)

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

13) 6cos2x3=0

14) sec2x2secx+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 f1(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: f1(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)=9x

20) f(x)=x2+3x+4

Answer
x32,f1(x)=32+124x7

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=f1(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)=82.567.5cos[(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)=82.567.5cos[(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.


This page titled 1.R: Chapter 1 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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