**Directions:**

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

Answer the following:

- 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?
- 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?
- 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?
- Simplify into one fraction:
- $x + \dfrac{3}{x}$
- $\dfrac{x+2}{x+1} + \dfrac{1}{x+3}$
- $\dfrac{1}{\sqrt{x}} + \dfrac{1}{\sqrt{x + h}}$
- $\sqrt{x+2} - \dfrac{1}{\sqrt{x+2}}$
- $\dfrac{x + 1}{\frac{x + 2}{x + 3}}$

- Simplify with the laws of exponents:
- $x^{\frac{2}{3}}\cdot x^3$
- $(x+1)^8(x+1)^5$
- $\dfrac{x(x + 3)^2 + x(x-1)}{x}$
- $(x^2 + 3x^6)^{3/4}(x^2 + 3x^6)^{\frac{1}{2}}$
- $(xy^2)^3$
- $(4x+3)^{-2}(x+1)$

- When finding the domain of a function, what are the two types of inputs we need to exclude?
- Suppose I have a function $f(x) = \dfrac{x}{(x+1)^2}$.
- Find $f(1), f(-a), f(x+h), f(x + h) - f(x)$ and fully expand each expression.

- Suppose a function takes an input $x$ and sends this to $\frac{1}{x + 1}$. What is the domain of $f(x)$?
- 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?
- Graph the function \[f(x) = \begin{cases}-x^2 & x \leq 3 \\ -x + 1 & x > 3\end{cases}\]
- Is the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ on the unit circle? Show using calculations.
- What is the reference number $\bar{t}$? What is the range of values $\bar{t}$ is allowed to be?
- Find the following terminal points associated with the following $t$ values:
- $t = 0$
- $t = \frac{3\pi}{2}$
- $t = \frac{2\pi}{3}$
- $t = \frac{5\pi}{6}$
- $t = -\frac{2\pi}{3}$
- $t = -\frac{9\pi}{4}$
- $t = 1000\pi + \frac{2\pi}{3}$
- $t = -83\pi + \frac{\pi}{3}$
- $t = \pi + 2\pi - 3\pi + \frac{4\pi}{3}$