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: $(5,\infty)$, Range: all real numbers

7.  $h\circ f$

8.  $g\circ f$

#### Solution

Domain: $(-\infty,-4) \cup (2,\infty)$, Range: all real numbers

Given the following functions, determine: $f(2), f(a)$ and $f(a+h)$

9. $f(x)=\frac{5}{2x-3}$

10. $f(x)=\frac{x}{x-1}$

#### Solution

$f(2) = 2, f(a) = \frac{a}{a-1}, f(a+h) = \frac{a+h}{a+h-1}$

11. $f(x)=\sqrt{7-x}$

12. $f(x)=\frac{x^2-1}{x-2}$

#### Solution

$f(2)$ does not exist since 2 is not in the domain of $f$, $f(a) = \frac{a^2-1}{a-2}, f(a+h) = \frac{(a+h)^2-1}{a+h-2} = \frac{a^2 + 2ah + h^2 -1}{a+h-2}$

Determine the domain of each function. Write your answer in interval notation.

13. $f(x)=\frac{x+5}{\sqrt{x^2+5x-14}}$

14. $f(x)=\frac{x-2}{\sqrt{x^2-3x+2}}$

#### Solution

Domain: $(-\infty,1) \cup (2,\infty)$

15. $f(x)=\sqrt{5x-1} - \sqrt{x}$

16. $f(x)=\sqrt{3x} - \sqrt{2-x}$

#### Solution

Domain: $[0,2]$

17. $f(x)=\frac{x^2+3}{\sqrt{x^2-9}}$

18. $f(x)=\frac{x^2+3}{\sqrt{9-x^2}}$

#### Solution

Domain: $(-3,3)$

19. $f(x)=e^{x^2+x-2}$

20. $f(x)= \frac{x}{1-e^{3x^2-4x-7}}$

#### Solution

Domain: $(-\infty,-1) \cup (-1, \frac{7}{3}) \cup (\frac{7}{3},\infty)$

21. $f(x)= \ln(16-x^2)$

22. $f(x)= \ln(2x^2-50)$

#### Solution

Domain: $(-\infty,-5) \cup (5,\infty)$

23. $f(x)= \frac{4x}{|x^2-36|}$

24. $f(x)= \frac{\sqrt{x-7}}{|x^2-49|}$

#### Solution

Domain: $(7,\infty)$

25. $f(x)= \frac{\sqrt{9-x}}{x^2-5x-24}$

26. $f(x)= \frac{\sqrt{x^2-6x+8}}{16-x^2}$

#### Solution

Domain: $(-\infty,-4) \cup (-4,2] \cup (4,\infty)$

27. $g(x) =\frac{\sqrt{4-x}}{x^2+3x+2}$

28. $h(x)=\dfrac{\sqrt{1-e^x}}{x+6}$

#### Solution

Domain: $(-\infty,-6)\cup(-6,0]$

Find the inverse of the following functions.

29. $f(x)=\frac{2x-1}{7}$

30. $g(x)=\frac{x-8}{2x+1}$

#### Solution

$g^{-1}(x)= \frac{-x-8}{2x-1}$

31. $g(x)=\ln(x-7)$

32. $f(x)=e^{2x}-3$

#### Solution

$f^{-1}(x)= \frac{lnx+3}{2}$

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

33.  $f(x)=2x^2+9x-5$

34.  $f(x)=x^3+2x^2-2x$

#### Solution

Degree of 3, $y$-intercept: 0, Zeros: 0, $\sqrt{3}-1, \, -1-\sqrt{3}$

Simplify the following trigonometric expressions.

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

36.  $\cos(2x)=\sin^2 x$

#### Solution

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

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

37.  $6\cos^2 x-3=0$

38.  $\sec^2 x-2\sec x+1=0$

#### Solution

$0, \, \pm 2\pi$

Solve the following logarithmic equations.

39.  $5^x=16$

40.  $\log_2 (x+4)=3$

#### 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.

41.  $f(x)=x^2+2x+1$

42.  $f(x)=\frac{1}{x}$

#### 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.

43.  $f(x)=\sqrt{9-x}$

44.  $f(x)=x^2+3x+4$

#### Solution

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

Sketch the following piece-wise functions.

45. $f(x)=\begin{cases} x+7, & x < 1 \\ x-5 & x \ge 1 \end{cases}$

46. $f(x)=\begin{cases} x^2-1, & x < -2 \\ x+2 & x \ge -2 \end{cases}$

#### Solution

47. $f(x)=\begin{cases} x-1, & x < 1 \\ 4, & x=1 \\ \sqrt{x+2} & x > 1 \end{cases}$

48. $f(x)=\begin{cases} x^2, & x < -1 \\ x-7, & x=-1 \\ \sqrt{x+3} & x > -1 \end{cases}$

#### Solution

Write the following absolute value functions as piece-wise functions and sketch.

49. $f(x)=|2x-5|$

50. $f(x)=|3x+7|$

#### Solution

$f(x)=\begin{cases} -(3x+7), & x < -\frac{7}{3} \\ 3x+7 & x \ge -\frac{7}{3} \end{cases}$

51.  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.

52.  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 53. 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.

54.  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

55.  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.

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

#### Solution

78.51%

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