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. [latex]f \circ g=g\circ f[/latex], assuming [latex]f[/latex] and [latex]g[/latex] 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
For the following problems, state the domain and range of the given functions:
[latex]f=x^2+2x-3,\phantom{\rule{3em}{0ex}}g=\ln(x-5),\phantom{\rule{3em}{0ex}}h=\frac{1}{x+4}[/latex]
5. [latex]h[/latex]
6. [latex]g[/latex]
Answer:
Domain: [latex]x>5[/latex], Range: all real numbers
7. [latex]h\circ f[/latex]
8. [latex]g\circ f[/latex]
Answer:
Domain: [latex]x>2[/latex] or [latex]x<-4[/latex], Range: all real numbers
Find the degree, [latex]y[/latex]-intercept, and zeros for the following polynomial functions.
9. [latex]f(x)=2x^2+9x-5[/latex]
10. [latex]f(x)=x^3+2x^2-2x[/latex]
Answer:
Degree of 3, [latex]y[/latex]-intercept: 0, Zeros: 0, [latex]\sqrt{3}-1, \, -1-\sqrt{3}[/latex]
Simplify the following trigonometric expressions.
11. [latex]\frac{\tan^2 x}{\sec^2 x}+\cos^2 x[/latex]
12. [latex] \cos(2x)=\sin^2 x[/latex]
Answer:
[latex] \cos(2x)[/latex] or [latex]\frac{1}{2}(\cos(2x)+1)[/latex]
Solve the following trigonometric equations on the interval [latex]\theta =[-2\pi ,2\pi][/latex] exactly.
13. [latex]6\cos^2 x-3=0[/latex]
14. [latex]\sec^2 x-2\sec x+1=0[/latex]
Answer:
[latex]0, \, \pm 2\pi [/latex]
Solve the following logarithmic equations.
15. [latex]5^x=16[/latex]
16. [latex]\log_2 (x+4)=3[/latex]
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 [latex]f^{-1}(x)[/latex] of the function. Justify your answer.
17. [latex]f(x)=x^2+2x+1[/latex]
18. [latex]f(x)=\frac{1}{x}[/latex]
Answer:
One-to-one; yes, the function has an inverse; inverse: [latex]f^{-1}(x)=\frac{1}{x}[/latex]
For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.
19. [latex]f(x)=\sqrt{9-x}[/latex]
20. [latex]f(x)=x^2+3x+4[/latex]
Answer:
[latex]x \ge -\frac{3}{2}, \, f^{-1}(x)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-7}[/latex]
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 [latex]55^{\circ}[/latex] 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 [latex]C=f(x)[/latex] 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. [latex]C(x)=300+7x[/latex] b. 100 shirts
23. a. Find the inverse function [latex]x=f^{-1}(C)[/latex] 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 [latex]P(t)=82.5-67.5\cos [(\pi /6)t][/latex], where [latex]t[/latex] is time in months ([latex]t=0[/latex] represents January 1) and [latex]P[/latex] 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 [latex]P(t)=82.5-67.5\cos [(\pi /6)t]+t[/latex], where [latex]t[/latex] is time in months ([latex]t=0[/latex] represents January 1) and [latex]P[/latex] 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 [latex]y=e^{rt}[/latex], where [latex]y[/latex] is the percentage of radiocarbon still present in the material, [latex]t[/latex] is the number of years passed, and [latex]r=-0.0001210[/latex] 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?
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