True or False? Justify your answers with a proof or a counterexample (1-4).
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.
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.
For the following problems (5-8), 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=\dfrac{1}{x+4}[/latex]
5. [latex]h[/latex]
6. [latex]g[/latex]
7. [latex]h\circ f[/latex]
8. [latex]g\circ f[/latex]
Find the degree, [latex]y[/latex]-intercept, and zeros for the following polynomial functions (9-10).
9. [latex]f(x)=2x^2+9x-5[/latex]
10. [latex]f(x)=x^3+2x^2-2x[/latex]
Simplify the following trigonometric expressions (11-12).
11. [latex]\dfrac{\tan^2 x}{\sec^2 x}+\cos^2 x[/latex]
12. [latex] \cos^{2} x=\sin^2 x[/latex]
Solve the following trigonometric equations (13-14) 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]
Solve the following logarithmic equations (15-16).
15. [latex]5^x=16[/latex]
16. [latex]\log_2 (x+4)=3[/latex]
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 answers (17-18).
17. [latex]f(x)=x^2+2x+1[/latex]
18. [latex]f(x)=\dfrac{1}{x}[/latex]
For the following problems (19-20), 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]
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 (22-23), 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.
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 (24-25), 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 \left[\left(\frac{\pi}{6}\right)t\right][/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?
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 \left[\left(\frac{\pi}{6}\right)t\right]+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 (26-27), 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?
27. Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?
