Homework 7

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Directions:

1. Show each step of your work and remember to 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.

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Answer the following:

1. State in mathematical terms what $f(x) = x$ is missing to transform into a general linear function $g(x) = ax + b$.
2. If I have a base function $f(x)$, explain in English what the following transformations will do:
   1. $f(x + 2)$
   2. $f(2x)$
   3. $-f(x)$
   4. $-2f(x - 3)$
   5. $-f(-x)$
3. Graph the following using transformations. Write out the blueprint of transformations to arrive at $f(x)$.
   1. $f(x) = \sqrt{x + 4} - 3$
   2. $f(x) = -(x-1)^2$
   3. $f(x) = 2 - 3(4x - 4)^2$
4. 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?
5. Suppose \[f(x) = x^3 \qquad g(x) = \left(\pi\left(x + 4\right)\right)^3 \qquad h(x) = \left(0.25x + 4\right)^3 \qquad i(x) =(x+4)^3\]
   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?
6. Suppose $f(x) = \sqrt{x}$. We want to vertically shift $f(x)$ upwards 3 units and left four units to make a new function $g(x)$. What is the formula of $g(x)$?
7. Starting from a base function $f(x)$, I want to create a function where I
   1. vertically shift up three units
   2. horizontally shift right four units
   3. vertical stretch of 2 units
   4. horizontal shrink of $\frac{1}{3}$ units
   5. reflection over the $x$-axis
   What should the formula of $f(x)$ look like?
8. For each red function, write out what the transformation from $f(x)$ should be:
9. Determine which functions are even/odd/neither. Use the definition of even and odd.
   1. $f(x) = x^2 - 1$
   2. $g(x) = \dfrac{1}{x}$
   3. $q(x) = x^5 + x^3 + x$
10. Suppose \[g(x) = \dfrac{1}{x-2} \qquad h(x) = \sqrt{x} \qquad i(x) = \dfrac{1}{x-3}\]
    1. Find $g(x)i(x)$ and it's domain.
    2. Find $h(x) + i(x)$ and it's domain.
11. Suppose \[g(x) = \sqrt{x - 2} \qquad h(x) = \dfrac{1}{x - 1} \qquad i(x) = \dfrac{1}{x-2}\]
    1. Find $g(x) + h(x)$ and it's domain.
    2. Find $1 - h(x)i(x)$ and it's domain.
12. If $f(x) = x^2 + 1$ and $g(x) = x - x^2$, find $f(x) - g(x)$ and $f(x)g(x)$.
13. Given the functions $f(x) = \sqrt{x - 2}$ and $g(x) = 5 - x^2$, find the following and fully simplify:
1. $f(g(0))$
2. $f\circ g$
3. $g \circ f$
4. $f \circ f$
5. $g \circ g$
14. Given the following functions $F(x)$, find two functions $f$ and $g$ where $f\circ g = F$. You are not allowed to choose $f(x) = x$ or $g(x) = x$.
    1. $F(x) = (x-3)^{-\frac{1}{2}}$
    2. $F(x) = \sqrt{x - 2} + \dfrac{1}{\sqrt{x - 2}}$
    3. $F(x) = \sqrt{(x^2 + 2x + 3)^3}$