# Homework 3

Directions: Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

1. For the curves that are actual functions in Problem 4 of Homework 2, which are one-to-one?
2. Suppose a function $f : \{1, 2\} \rightarrow \{1, 2\}$ is defined as such: $f(1) = 2, f(2) = 1$.
1. State the domain and range of this function.
2. Is this function one-to-one?
3. If it is one-to-one, state the behavior of $f^{-1}$.
3. Suppose $f(x)$ has domain $[1, 5]$ and range $(-\infty, \infty)$. Further suppose $f(x)$ is one-to-one.
1. Does $f^{-1}$ exist?
2. If $f^{-1}$ exists, what is the domain and range of $f^{-1}$?
4. Draw a graph of a one-to-one function.
5. Algebraically find the inverse of $f(x) = x$, if you can.
6. Algebraically find the inverse of $f(x) = \dfrac{x + 2}{2x + 3}$, if you can.
7. Algebraically find the inverse of $f(x) = \sqrt{x - 3}, x \leq 12$ if you can.
8. Algebraically find the inverse of $f(x) = \dfrac{(x+2)(x+1)}{(x+2)(x + 3)}$, if you can.
1. Hint: The domain restriction is $\{x : x \neq -2 \text{ and } x \neq -3\}$, even if you cancel. Keep domain restrictions of the original function!
9. Algebraically find the inverse of $f(x) = \dfrac{x}{x+1} + \dfrac{2x + 1}{x + 1}$, if you can.
10. Suppose $f(x) = x$ and $g(x) = \dfrac{1}{x}$. Are $f(x)$ and $g(x)$ inverses? Show using the Inverse Function Property.
11. Suppose $f(x) = x^2$ and $g(x) = \sqrt{x}$. Are $f(x)$ and $g(x)$ inverses? Show using the Inverse Function Property.
12. Is the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ on the unit circle? Show using calculations.
13. What is the reference number $\bar{t}$? What is the range of values $\bar{t}$ is allowed to be?
14. Find the following terminal points associated with the following $t$ values:
1. $t = 0$
2. $t = \frac{3\pi}{2}$
3. $t = \frac{2\pi}{3}$
4. $t = \frac{5\pi}{6}$
5. $t = -\frac{2\pi}{3}$
6. $t = -\frac{9\pi}{4}$
7. $t = 1000\pi + \frac{2\pi}{3}$
8. $t = -83\pi + \frac{\pi}{3}$
9. $t = \pi + 2\pi - 3\pi + \frac{4\pi}{3}$
15. (skip) Find the six trigonmetric functions of $t = \frac{2\pi}{3}$.
16. (skip) Find the six trigonmetric functions of $t = -\frac{5\pi}{6}$.
17. (skip) Why is $\tan\left(\frac{\pi}{2}\right)$ undefined?