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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.

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.

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. $g$

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

7. $h\circ f$

8. $g\circ f$

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. $f(x)=x^3+2x^2-2x$

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. $\cos(2x)=\sin^2 x$

$\cos(2x)$ 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. $\sec^2 x-2\sec x+1=0$

$0, \, \pm 2\pi$

Solve the following logarithmic equations.

15. $5^x=16$

16. $\log_2 (x+4)=3$

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. $f(x)=\frac{1}{x}$

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. $f(x)=x^2+3x+4$

$x \ge -\frac{3}{2}, \, f^{-1}(x)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-7}$

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. 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 $P(t)=82.5-67.5\cos [(\pi /6)t]$, 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?

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?