Homework 7

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:

Consider $y = f(x)$, where $x$ is time and $y$ is temperature. If you find the inverse $f^{-1}(x)$, what does the $x$ in the inverse mean?
Suppose $f$ is not a one-to-one function. Explain what property is violated when we try to define $f^{-1}$.
Are $f(x)$ and $g(x)$ inverses of each other? Show using the Inverse Function Property.
1. $f(x) = x^2, \qquad g(x) = \sqrt{x}$
2. $f(x) = 2 - 5x, \qquad g(x) = \dfrac{2 - x}{5}$
3. $f(x) = \dfrac{1}{x-1}, \qquad g(x) = \dfrac{1 + x}{x}$
  Hint This is a compound fraction. I advise evaluating $f\circ g$ so you only need to deal with the denominator.
The function $f(x) = x^2$ is not one-to-one. What could we restrict the domain to to make $f(x)$ one-to-one?
Hint Look at the graph. Which part can you delete?
Find the codomain of the following functions.
1. $f(x) = \sqrt{x}$
2. $f(x) = x^2 - 77$
  Hint Transformations.
3. $f(x) = \lvert x \rvert + 1$
Find the inverse function of each of the following functions or explain why it is not possible.
Hint Check one-to-one first. Maybe inverse doesn't exist.
1. $f(x) = 3x + 5$
2. $f(x) = (x-1)^2$
3. $f(x) = \dfrac{x}{x + 2}$
4. $f(x) = 4-x^2, \qquad x \geq 0$
5. $f(x) = \sqrt{4 - x^2}, \qquad 0 \leq x \leq 2$
6. $f(x) = \lvert 3x - 6 \rvert, \qquad x \geq 0$
7. $f(x) = 2 + \sqrt{x + 3}$
  Hint Similar technique to solving root equations.
8. $f(x) = \dfrac{2x + 4}{x - 7}$
Given this graph of a function: find
1. $f^{-1}(2)$
2. $f^{-1}(5)$
3. $f^{-1}(6)$
Hint If $f(x) = y$, then $f^{-1}(y) = x$. What is the $y$?
Complete the square for the following expressions:
1. $f(x) = x^2 - 6x$
2. $f(x) = x^2 - 4x + 1$
3. $f(x) = -2x^2 - 8x$
4. $f(x) = -3x^2 + 6x$
Express the following quadratics in standard form and sketch a rough graph:
1. $f(x) = x^2 - 6x$
2. $f(x) = 3x^2 - 6x + 1$
3. $f(x) = x^2 - 8x + 8$
4. $g(x) = 2x^2 + 8x + 11$
Give an example of a polynomial with degree $6$, leading coefficient $-3$, and constant coefficient $-2$.
Consider a polynomial $P(x)$.
1. If $(x+3)$ is a factor of $P(x)$, is $-3$ a zero of $P(x)$?
2. If $P(-2) = 0$, is $(x-2)$ a factor of $P(x)$?
3. If $x = 2$ is a $x$-intercept of $P(x)$, is $(x-2)$ a factor of $P(x)$?
4. If $c$ is a zero of $P(x)$, is $(x+c)$ a factor of $P(x)$?
5. If $(x + 1)$ is a factor of $P(x)$, is $x = 1$ a $x$-intercept of $P(x)$?
6. If $P(3) = 0$, is $x = 3$ a $x$-intercept of $P(x)$?
Suppose $P(x)$ is a polynomial. What is the domain of $P(x)$?
Suppose $P(x)$ has end behavior $y\rightarrow -\infty$ as $x \rightarrow \infty$ and $y\rightarrow -\infty$ as $x\rightarrow -\infty$. Describe three possible leading terms for $P(x)$.
Suppose $P(x)$ has end behavior $y\rightarrow \infty$ as $x \rightarrow \infty$ and $y\rightarrow -\infty$ as $x\rightarrow -\infty$. Describe three possible leading terms for $P(x)$.
Determine the leading term for the following polynomials:
1. $P(x) = 2x(x-2)(x-3)^2$
2. $P(x) = -(x-1)(-2x+1)$
3. $P(x) = (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)^2$
4. $P(x) = -(x-3)^{100}(x-2)^{200}(2x - 1)$
Suppose you're instructed to "Draw a graph of...". What test does the graph need to pass and why?
Suppose $P(x)$ has zeros $-2$ with multiplicity $3$ and $4$ with multiplicity $2$. Write down a possible algebraic formula for $P(x)$.
Which graph(s) are graphs of polynomials?
Graph the following polynomials by hand using the four step process. Make sure the zeroes, end behavior, and zero multiplicity are displayed correctly.
You must show the intermediate work in each step for full credit.
1. $P(x) = (x - 1)(x + 2)$
2. $P(x) = -x(x-3)(2x+2)$
3. $P(x) = (x-1)^2(x+2)^3$
4. $P(x) = x^3(x+2)(x-3)^2$
5. $P(x) = x^4 - 2x^3 - 8x + 16$
6. $P(x) = x^3 - x^2 - 6x$
7. $P(x) = x^5 - 9x^3$
8. $P(x) = 2x^3 - 2x^2 - 12x$
9. $P(x) = -2x^3(2x-4)^2(x+2)^2$

Section 3.3 problems will be on next week's homework.