# Homework 1

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. Suppose there is a variable $x$ which is associated with a value $f(x)$. I find that two different inputs give the same evaluation. In particular, I find $x = -2$ and $x = 2$ have $f(-2) = f(2)$. Is $f(x)$ a function?
2. Suppose I have an expression $f(x)$ and I find one input which gives two different evaluations. In particular, I find that $x = 2$ spits out $f(2) = 5$ and $f(2) = 3$. Is $f$ a function?
3. Suppose there is an expression $f(x) = \dfrac{(a + h)(b+h) + x}{(a+h)^3 - 5}$ Am I allowed to cancel out the $a+h$'s to get $\dfrac{x}{(a+h)^2 - 5}$ If I am not, when am I allowed to cancel in this fashion?
4. Simplify into one fraction:
1. $x + \dfrac{3}{x}$
2. $\dfrac{x+2}{x+1} + \dfrac{1}{x+3}$
3. $\dfrac{1}{\sqrt{x}} + \dfrac{1}{\sqrt{x + h}}$
4. $\sqrt{x+2} - \dfrac{1}{\sqrt{x+2}}$
5. $\dfrac{x + 1}{\frac{x + 2}{x + 3}}$
5. Simplify with the laws of exponents:
1. $x^{\frac{2}{3}}\cdot x^3$
2. $(x+1)^8(x+1)^5$
3. $\dfrac{x(x + 3)^2 + x(x-1)}{x}$
4. $(x^2 + 3x^6)^{3/4}(x^2 + 3x^6)^{\frac{1}{2}}$
5. $(xy^2)^3$
6. $(4x+3)^{-2}(x+1)$
6. When finding the domain of a function, what are the two types of inputs we need to exclude?
7. Suppose I have a function $f(x) = \dfrac{x}{(x+1)^2}$.
1. Find $f(1), f(-a), f(x+h), f(x + h) - f(x)$ and fully expand each expression.
8. Suppose a function takes an input $x$ and sends this to $\frac{1}{x + 1}$. What is the domain of $f(x)$?
9. Draw a coordinate plane and graph the functions $f(x) = x^2, g(x) = x^4$ and $h(x) = x^6$. What is similar between the graphs?
10. Graph the function $f(x) = \begin{cases}-x^2 & x \leq 3 \\ -x + 1 & x > 3\end{cases}$
11. Is the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ on the unit circle? Show using calculations.
12. What is the reference number $\bar{t}$? What is the range of values $\bar{t}$ is allowed to be?
13. 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. $t = \pi + 2\pi - 3\pi + \frac{4\pi}{3}$