# Homework 2

Directions:

1. Show each step of your work and fully simplify each expression.
2. Turn in your answers in class on a physical piece of paper.
3. Staple multiple sheets together.
4. Feel free to use Desmos for graphing.

1. Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
2. Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
3. Why is $\tan\left(\frac{\pi}{2}\right)$ undefined?
4. Suppose $f(x) = x^2 + 2 \qquad g(x) = \dfrac{x + 1}{x - 1}$
1. Find $f\circ g, g\circ f, (f\circ g)(5), (g\circ f)(5)$.
5. For the next few parts, find $\dfrac{f(a + h) - f(a)}{h},\qquad h \neq 0$ and simplify fully:
1. $f(x) = x^2$
2. $f(x) = \dfrac{1}{x}$
3. $f(x) = x^3 - 1$
6. For the next few parts, find two functions $f$ and $g$ such that $h = f\circ g$. You are not allowed to choose $f(x) = x$ or $g(x) = x$.
1. $h(x) = (x^3 + x + 1)^3$
2. $h(x) = \dfrac{1}{\sqrt{x^2 - 2}}$
3. $h(x) = \dfrac{1}{\sqrt{x + 1}} + \sqrt{x + 1}$
7. Use the law of exponents to fully simplify $\dfrac{\sqrt{a\sqrt{b}}}{\sqrt{ab}}$
8. Solve each equation for $x$:
1. $\ln(x^2 - 1) = 3$
2. $e^{2x}-5e^x + 4 = 0$
3. $2^{x-5}=3$
9. If $f$ is a one-to-one function and $f(6) = 17$, what is $f^{-1}(17)$?
10. Show why $f^{-1}$ is not the same function as $\dfrac{1}{f}$ with an example. Hint: The function $f(x) = x^n$
11. If $f(x) = x^2, x > 0$, find $f(f^{-1}(2))$.
12. Combine into one logarithm:
1. $2 \ln 5 + 3 \ln 2$
2. $\ln(a + b) - \ln(a-b) - 2\ln c$
13. List the first five terms of these sequences. Make sure you have a list of numbers.
1. $a_n = \dfrac{2n}{n^2 + 1}$
2. $a_n = \cos\dfrac{n\pi}{2}$
14. Find a formula for $a_n$ for the following sequences:
1. $1, 0, -1,0, 1, 0, -1, 0$
2. $\dfrac{1}{2}, -\dfrac{4}{3}, \dfrac{9}{4}, - \dfrac{16}{5}, \dfrac{25}{6}, \dots$