1. Functions and Graphs

Chapter 1 Review Exercises

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

1. A function is always one-to-one.

2. f \circ g=g\circ 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,\phantom{\rule{3em}{0ex}}g=\ln(x-5),\phantom{\rule{3em}{0ex}}h=\frac{1}{x+4}

5. h

6. g

Solution

Domain: x>5, Range: all real numbers

7. h\circ f

8. g\circ f

Solution

Domain: x>2 or x<-4, Range: all real numbers

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

9. f(x)=2x^2+9x-5

10. 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.

11. \frac{\tan^2 x}{\sec^2 x}+\cos^2 x

12.  \cos(2x)=\sin^2 x

Solution

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

Solve the following trigonometric equations on the interval \theta =[-2\pi ,2\pi] exactly.

13. 6\cos^2 x-3=0

14. \sec^2 x-2\sec x+1=0

Solution

0, \, \pm 2\pi

Solve the following logarithmic equations.

15. 5^x=16

16. \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.

17. f(x)=x^2+2x+1

18. f(x)=\frac{1}{x}

Solution

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

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)=\sqrt{9-x}

20. f(x)=x^2+3x+4

Solution

x \ge -\frac{3}{2}, \, f^{-1}(x)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-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^{\circ} 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.

Solution

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)=82.5-67.5\cos [(\pi /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

25. In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as P(t)=82.5-67.5\cos [(\pi /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 decay rate of radiocarbon.

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

Solution

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?

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