# Homework 2

Directions: Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

1. If $f(x) = \dfrac{x}{x+1}$, find $f(a + h)$.
2. Suppose $f(x) = x^2 + 3$. What is the physical meaning of $f(3)$ on the coordinate plane?
3. Suppose $f(x) = x^2$ and $g(x) = x^3$. In class we showed the range of $f(x)$ is $[0, \infty)$.
1. What is the range of $g(x)$?
2. Why is the range of $g(x)$ different than $f(x)$?
4. Which of the four curves below are actually functions? State their colors.
5. What is the domain and range of the following function?
6. If $f(x) = x$ and $g(x) = 2(x - 3)$, what two transformations need to be applied to $f(x)$ to end up at $g(x)$? Be sure to state the value of each transformation.
7. Suppose $f(x) = x^2$ and $g(x) = -2\left[\frac{1}{3}\left(x-2\right)\right]^2 + 3$. State the order of transformations that $f(x)$ needs to undergo to end up at $g(x)$.
8. Suppose $f(x) = \sqrt{x}$ and $g(x) = \sqrt{-\frac{2}{7}\left(x+3\right)} + 4$. State the order of transformations that $f(x)$ needs to undergo to end up at $g(x)$.
9. Given $f(x)$ and a new function $g(x) = -f(x) + 3$, do we perform the reflection around the $x$-axis first or the vertical shift? Why?
10. If $f(x)$ is an even function, does that mean all of the coefficients of $f(x)$ are even? For example, $f(x) = 2x^3 + 4$ would be even because $2$ and $4$ are even.
11. Suppose $f(x)$ is a function that takes $x$ to the fourth power, then adds $x$ to the second power.
1. Write down the algebraic form of $f(x)$.
2. Is $f(x)$ even, odd, or neither?
12. Given the functions $f(x) = \sqrt{x - 2}$ and $g(x) = 5 - x^2$, find the following:
1. $f(g(0))$
2. $f\circ g$
3. $f \circ f$
13. Given the function $F(x) = \dfrac{1}{\sqrt{x - 3}}$, write $F(x)$ as a composition of functions $f$ and $g$ (meaning determine $f$ and $g$ such that $F(x) = (f\circ g)(x)$).
14. Suppose $f(x) = x \qquad g(x) = \dfrac{1}{x-2} \qquad h(x) = \sqrt{x} \qquad i(x) = \dfrac{1}{x-3}$
1. Find $f(x) + g(x) - h(x) - i(x)$ and it's domain.
2. Find $f(x) \cdot g(x) \cdot h(x) \cdot i(x)$ and it's domain.
15. Suppose $f(x) = x^2 \qquad g(x) = \sqrt{x} \qquad h(x) = \dfrac{1}{x - 1} \qquad i(x) = \dfrac{1}{x-2}$
1. Find $f(x) + g(x) - h(x) - i(x)$ and it's domain.
2. Find $\dfrac{f(x) \cdot g(x) \cdot h(x)}{i(x)}$ and it's domain.
16. Suppose $f(x) = \sqrt{x} \qquad g(x) = \sqrt{x - 2} \qquad h(x) = \sqrt{\frac{1}{2}x - 2} \qquad i(x) = \sqrt{\frac{1}{2}\left(x-2\right)}$
1. Is $g(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
2. Is $h(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
3. Is $i(x)$ horizontally shifted two units to the right from $f(x)$? Why or why not?
17. Is $f(x) = \dfrac{1}{x}$ even, odd, or neither?
18. Given $f(x) = \dfrac{x^2 + 1}{\sqrt{x}}$, evaluate the following:
1. $f(0)$
2. $f(1)$
3. $f(a)$
4. $f(a^2)$
5. $f(x + 1)$
6. $f(a\cdot(x+1))$
19. Suppose $f(x) = x^3 \qquad g(x) = \left(\pi\left(x + 4\right)\right)^4 \qquad h(x) = \left(0.25x + 4\right)^3 \qquad i(x) =(x+4)^3$
1. Is $g(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
2. Is $h(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?
3. Is $i(x)$ horizontally shifted four units to the left from $f(x)$? Why or why not?