# Homework 1

Directions: Turn in your answers in class on a physical piece of paper.

1. Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$. Is the domain of $f(x) = \sqrt{x}$ all real numbers?
2. In the expression $\dfrac{x}{x + 1}$, can we cancel out the $x$'s to get $\frac{0}{0 + 1} = 0$? If we cannot, explain why.
3. For the expression $\dfrac{x + 1}{x}$, can we cancel out the $x$'s to get $\frac{0 + 1}{0}$ which is undefined? If we cannot, explain why.
4. In the expression $\dfrac{x + 1}{x}$, can we rewrite this as $\dfrac{x}{x} + \dfrac{1}{x}$, which simplifies as $1 + \dfrac{1}{x}$? If we cannot, explain why.
5. 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?
6. 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?
7. Suppose I have an expression $x(y^2 + y)$ and I am asked to expand it. I try $x\cdot y^2 + x \cdot y = xy^2 + xy$. What property did I use?
8. What factors do $4$ and $6$ have in common?
9. What factors do $(x+1)^2(x+3)$ and $(x+1)(x+2)(x+3)^3$ have in common?
10. What factors do $xy^2$ and $xy$ have in common?
11. Given the previous problem, what do we factor out of $xy^2 + xy$?
12. What is the reverse process of expanding an expression?
13. A student tries to simplify $x^2 + x^3$ by applying exponent laws. They write $x^2 + x^3 = x^{2+3} = x^5$ Why is this incorrect?
14. A student tries to simplify $x^2 \cdot x^3$ by applying exponent laws. They write $x^2 \cdot x^3 = x^{2\cdot3} = x^6$ Why is this incorrect?
15. A student tries to simplify $(a + b)^2$ by applying exponent laws. They write $(a + b)^2 = a^2 + b^2$ Why is this incorrect?
16. A student tries to simplify $\frac{3x}{x^2 + 3x + 1}$ by cancelling common terms. Noticing the $3x$ is shared, they write $\frac{3x}{x^2 + 3x + 1} =\frac{1}{x^2 + 1}$ Why is this incorrect?
17. (skip this) Suppose I have a function $f(x)$ and $g(x) = A + Bf(Cx + D)$. State which parts of $g(x)$ correspond to the following transformations of $f(x)$:
• Vertical shift
• Horizontal shift
• Vertical stretch/shrink
• Horizontal stretch/shrink
• Reflection around $x$-axis
• Reflection around $y$-axis

1. $\dfrac{1}{x+1} + \dfrac{1}{(x+1)^2}$
2. $\dfrac{1}{(x+1)(x+2)} + \dfrac{1}{(x+1)^2}$
3. $\dfrac{1}{x} + \dfrac{1}{x^2} + \dfrac{1}{x^3}$
1. $\dfrac{2}{4}$
2. $\dfrac{(x + 2)}{2(x+2)}$
3. $\dfrac{(x+1)(x + 2)}{2(x+3)(x+2)^2}$