**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:

- Rationalize (Lecture Note IV) the numerator for the expression \[\dfrac{\sqrt{x + h} - \sqrt{x}}{h}\]
- Rationalize the denominator for the expression \[\dfrac{16x - x^2}{4 - \sqrt{x}}\]
- Rationalize the denominator for the expression \[\dfrac{x^2}{\sqrt{x^2 + 9} - 3}\]
- Rationalize the numerator for the expression \[\dfrac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}\]
- Suppose your professor tells you to simplify the expression \[\dfrac{\dfrac{2}{x-2} - \dfrac{3}{x - 3}}{1 + \dfrac{1}{x + 4}}\]
- What type of expression is this called?
- How should you think about approaching this type of expression?

- Perform the indicated operation and fully simplify (meaning write as one fraction only).

Get rid of all negative exponents.- $\dfrac{3}{10} + \dfrac{4}{15}$
- $\dfrac{3}{5} \div \dfrac{6}{5}$
- $2 \cdot \dfrac{1}{3} \cdot \dfrac{5}{7}$
- $\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
- $\dfrac{1 + \dfrac{1}{x}}{\dfrac{1}{x} - 2}$
- $\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
- $\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x-1}$
- $\dfrac{x^{-1} + y^{-1}}{4} \ \ $

Hint: Use definition of negative exponent to see this is a compound fraction. - $\dfrac{\dfrac{1}{\sqrt{x+h}} - \dfrac{1}{\sqrt{x}}}{h} \ \ $

Hint: This is a compound fraction. Focusing on the numerator as a subproblem:- Subtract the fractions (another hint: Problem 4).
- Rationalize the numerator.
- Divide.
- Cancel the global factor $h$.

- $\dfrac{4(x+3)(x-1)}{2(x-1)}$
- $\dfrac{x^2 + 2x + 1}{x^2 - 1}$
- $\dfrac{x^2h + 2xh + h}{h}$
- $2 + \dfrac{1}{x + 3}$
- $\dfrac{(x+h)^2 - x^2}{h}$
- $\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}$
- $\dfrac{\dfrac{1}{x + h} - \dfrac{1}{x}}{h}$

- Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
- Suppose I have an expression $f(x)$ and I find two different inputs that give the same evaluation. In particular, I find that $x = 2$ and $x = 3$ gives $f(2) = f(3)$. Is $f$ 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?
- Write down two examples of functions which you encounter in your daily life.
- Let $f(x) = x^2 - x + 1$. Evaluate and
**simplify**the following:- $f(1)$
- $f(a)$
- $f(-a)$
- $f(x + h)$
- $f(x + h) - f(x)$
- $f(-x^2)$

- Let $f(x) = \dfrac{1}{x + 1}$ Evaluate and
**simplify**the following:- $f(-1)$
- $f(x + h)$
- $f(x + h) - f(x)$

- Let \[f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}\] Evaluate the following:
- $f(0)$
- $f(1)$
- $f(2)$
- $f(3)$

- Draw a rough sketch of the previous graph by hand. You may verify your graph in Desmos.

Hint: To graph a piecewise function in Desmos, you need to specify a domain restriction.

You can graph $3x + 2, x \geq 2$ by typing the symbols $3x + 2 \{x \geq 2\}$ into Desmos, where the $\geq$ sign is created by first typing $>$ then $=$. - Graph the function by hand: \[f(x) = \begin{cases} -x^2 & x > 0 \\ -x - 1 & x \leq 0 \end{cases}\]
- Graph the function by hand: \[g(x) = \begin{cases} \sin(x) & x > 0 \\ x & x < 0 \\ 3 & x = 0\end{cases}\]
- Draw one curve in the plane that isn't the graph of a function.
- Read these two lecture notes on trigonometric functions: 5.1: Unit Circle and 5.2: Trigonometric Functions. This is mandatory reading if you do not remember trigonometric functions.

Write "Without using the internet, I can evaluate trigonometric functions and know their graph shapes." to get credit for this problem. - For what inputs $x$ does the function $f(x) = \cos(x)$ output $0$? (Hint: 5.1 + 5.2 Lecture notes. In other words, which radian value results in an $x$-coordinate of 0 on the unit circle?).
- Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
- Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
- Suppose \[f(x) = \sin(x) + \cos(x) \qquad g(x) = \tan(x)\]
Evaluate and simplify the following expressions without a calculator/the internet:
- $f(0)$
- $g(0)$
- $f\left(\frac{\pi}{2}\right) + g(\pi)$
- $f(x)\cdot g(x)$
- $f(x) - g(x)$

- Redefine the following rational functions with their domain exclusions (holes):
- $f(x) = \dfrac{x^2 - 1}{x-1}$
- $f(x) = \dfrac{2x^2 + 3x - 2}{x + 2}$
- $f(x) = \dfrac{x^3 - 3x^2 - 4x}{x}$

- Suppose \[f(x) = \sin(x) \qquad g(x) = x^2 - x \qquad h(x) = f(x)g(x)\] Evaluate the following:
- $f\circ g$
- $g \circ f$
- $h \circ f$

- Suppose $f(x)$ and $g(x)$ are two different functions. In the expression $f(x)g(x)$, is $f(x)$ a term or a factor in the global context?
- Suppose \[f(x) = x^2 - x \qquad g(x) = x^3 - x^2 + 1\]
Evaluate (and always remember to simplify) the following:
- $f(x)g(x)$
- $f(x)g(x) - [f(x)]^2$
- $\dfrac{f(x)[g(x)]^3}{[g(x)]^2}$

Hint: Could you use the previous problem's idea to perhaps simplify your calculation?

- You are given a function $F(x)$. Find two functions $f, g$ where $F = f \circ g$.
- $F(x) = \sin(\cos x)$
- $F(x) = \sin^2(x)$
- $F(x) = \sin(x^2)$
- $F(x) = (x^3 - x^2 - 1)^{2/3}$
- $F(x) = (x^2 - x)^2$
- $F(x) = \sqrt[5]{(x + 1)^3}$
- $F(x) = \sec(\tan(x))$

- Find the domain of the following functions:
- $f(x) = \dfrac{1}{x^2 - 1}$
- $f(x) = \sqrt{x} + \dfrac{1}{x}$
- $f(x) = \tan(x)$
- $f(x) = \sin(x)$
- $f(x) = \dfrac{1}{\sqrt{x}}$

Hint: Both problems are present.

- Suppose $f(x) = mx + b$ where $m$ and $b$ are real numbers. What does the graph of $f(x)$ look like?