# 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. Rationalize (Lecture Note IV) the numerator for the expression $\dfrac{\sqrt{x + h} - \sqrt{x}}{h}$
2. Rationalize the denominator for the expression $\dfrac{16x - x^2}{4 - \sqrt{x}}$
3. Rationalize the denominator for the expression $\dfrac{x^2}{\sqrt{x^2 + 9} - 3}$
4. Rationalize the numerator for the expression $\dfrac{\sqrt{x} - \sqrt{x+h}}{\sqrt{x}\sqrt{x+h}}$
5. Suppose your professor tells you to simplify the expression $\dfrac{\dfrac{2}{x-2} - \dfrac{3}{x - 3}}{1 + \dfrac{1}{x + 4}}$
1. What type of expression is this called?
2. How should you think about approaching this type of expression?
6. Perform the indicated operation and fully simplify (meaning write as one fraction only).
Get rid of all negative exponents.
1. $\dfrac{3}{10} + \dfrac{4}{15}$
2. $\dfrac{3}{5} \div \dfrac{6}{5}$
3. $2 \cdot \dfrac{1}{3} \cdot \dfrac{5}{7}$
4. $\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
5. $\dfrac{1 + \dfrac{1}{x}}{\dfrac{1}{x} - 2}$
6. $\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
7. $\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x-1}$
8. $\dfrac{x^{-1} + y^{-1}}{4} \ \$
Hint: Use definition of negative exponent to see this is a compound fraction.
9. $\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 $h$.
10. $\dfrac{4(x+3)(x-1)}{2(x-1)}$
11. $\dfrac{x^2 + 2x + 1}{x^2 - 1}$
12. $\dfrac{x^2h + 2xh + h}{h}$
13. $2 + \dfrac{1}{x + 3}$
14. $\dfrac{(x+h)^2 - x^2}{h}$
15. $\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}$
16. $\dfrac{\dfrac{1}{x + h} - \dfrac{1}{x}}{h}$
7. Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
8. 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?
9. 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?
10. Write down two examples of functions which you encounter in your daily life.
11. Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
1. $f(1)$
2. $f(a)$
3. $f(-a)$
4. $f(x + h)$
5. $f(x + h) - f(x)$
6. $f(-x^2)$
12. Let $f(x) = \dfrac{1}{x + 1}$ Evaluate and simplify the following:
1. $f(-1)$
2. $f(x + h)$
3. $f(x + h) - f(x)$
13. Let $f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}$ Evaluate the following:
1. $f(0)$
2. $f(1)$
3. $f(2)$
4. $f(3)$
14. 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 $=$.
15. Graph the function by hand: $f(x) = \begin{cases} -x^2 & x > 0 \\ -x - 1 & x \leq 0 \end{cases}$
16. Graph the function by hand: $g(x) = \begin{cases} \sin(x) & x > 0 \\ x & x < 0 \\ 3 & x = 0\end{cases}$
17. Draw one curve in the plane that isn't the graph of a function.
18. 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.
19. 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?).
20. Find the six trigonometric functions of $t = \frac{2\pi}{3}$.
21. Find the six trigonometric functions of $t = -\frac{5\pi}{6}$.
22. Suppose $f(x) = \sin(x) + \cos(x) \qquad g(x) = \tan(x)$ Evaluate and simplify the following expressions without a calculator/the internet:
1. $f(0)$
2. $g(0)$
3. $f\left(\frac{\pi}{2}\right) + g(\pi)$
4. $f(x)\cdot g(x)$
5. $f(x) - g(x)$
23. Redefine the following rational functions with their domain exclusions (holes):
1. $f(x) = \dfrac{x^2 - 1}{x-1}$
2. $f(x) = \dfrac{2x^2 + 3x - 2}{x + 2}$
3. $f(x) = \dfrac{x^3 - 3x^2 - 4x}{x}$
24. Suppose $f(x) = \sin(x) \qquad g(x) = x^2 - x \qquad h(x) = f(x)g(x)$ Evaluate the following:
1. $f\circ g$
2. $g \circ f$
3. $h \circ f$
25. 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?
26. Suppose $f(x) = x^2 - x \qquad g(x) = x^3 - x^2 + 1$ Evaluate (and always remember to simplify) the following:
1. $f(x)g(x)$
2. $f(x)g(x) - [f(x)]^2$
3. $\dfrac{f(x)[g(x)]^3}{[g(x)]^2}$
Hint: Could you use the previous problem's idea to perhaps simplify your calculation?
27. You are given a function $F(x)$. Find two functions $f, g$ where $F = f \circ g$.
1. $F(x) = \sin(\cos x)$
2. $F(x) = \sin^2(x)$
3. $F(x) = \sin(x^2)$
4. $F(x) = (x^3 - x^2 - 1)^{2/3}$
5. $F(x) = (x^2 - x)^2$
6. $F(x) = \sqrt[5]{(x + 1)^3}$
7. $F(x) = \sec(\tan(x))$
28. Find the domain of the following functions:
1. $f(x) = \dfrac{1}{x^2 - 1}$
2. $f(x) = \sqrt{x} + \dfrac{1}{x}$
3. $f(x) = \tan(x)$
4. $f(x) = \sin(x)$
5. $f(x) = \dfrac{1}{\sqrt{x}}$
Hint: Both problems are present.
29. Suppose $f(x) = mx + b$ where $m$ and $b$ are real numbers. What does the graph of $f(x)$ look like?