$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

1R: Chapter 1 Review Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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.

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

State the domain and range of the given functions:

$$f=x^2+2x−3$$, $$g=\ln(x−5)$$, $$h=\dfrac{1}{x+4}$$

5) h

6) g

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

7) $$h∘f$$

8) $$g∘f$$

Domain: $$x>2$$ and $$x<−4$$, Range: all real numbers

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

9) $$f(x)=2x^2+9x−5$$

10) $$f(x)=x^3+2x^2−2x$$

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

Simplify the following trigonometric expressions.

11) $$\dfrac{\tan^2x}{\sec^2x}+{\cos^2x}$$

12) $$\cos^2x-\sin^2x$$

$$\cos(2x)$$

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

13) $$6\cos 2x−3=0$$

14) $$\sec^2x−2\sec x+1=0$$

$$0,±2π$$

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) $$f(x)=\dfrac{1}{x}$$

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

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) $$f(x)=x^2+3x+4$$

$$x≥−\frac{3}{2},\quad f^{−1}(x)=−\frac{3}{2}+\frac{1}{2}\sqrt{4x−7}$$

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 $$P(t)=82.5−67.5\cos[(π/6)t]$$, 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?

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 $$P(t)=82.5−67.5\cos[(π/6)t]+t$$, 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 $$y=e^{rt}$$, 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?

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?