Homework 5

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.

Answer the following:

  1. Suppose your friend was given a function \[f(x) = x^2 - 3\] and they are asked to find $f(x + h) - f(x)$. Your friend ends up writing \[f(x + h) - f(x) = x^2 - 3 + h - x^2 - 3\] Identify the two mistakes that were made.
  2. Find the domain of the following functions:
    1. $f(x) = \dfrac{1}{\sqrt{2x - 1}}$
    2. $f(x) = \dfrac{1}{x\sqrt{2x - 4}}$
      Hint Use the zero product property when setting denominator to zero.
    3. $g(x) = \sqrt{x-2} - \dfrac{2x}{\sqrt{-x + 3}}$
  3. 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. $f(3)$
    5. $f(4)$
  4. Let \[f(x) = \begin{cases}-1 & x < -1 \\ x & -1 \leq x \leq 1 \\ -1 & x > 1\end{cases}\] Evaluate the following:
    1. $f(-2)$
    2. $f(-1)$
    3. $f(0)$
    4. $f(0.99)$
    5. $f(1)$
    6. $f(2)$
  5. Sketch a graph on the coordinate plane for the following functions using the table. Feel free to verify your graphs with Desmos.
    1. $f(x) = 1-x^2$
    2. $g(x) = x^3 + 1$
    3. $f(x) = \begin{cases} -x - 1 & x \leq 1 \\ x + 1 & x > 1\end{cases}$
    4. $f(x) = \begin{cases} x & x > 1 \\ x^2 - 1 & x \leq -1\end{cases}$
    5. $f(x) = \begin{cases} 1 & x \geq 2 \\ -1 & x < 2 \end{cases}$
    6. $g(x) = \begin{cases} x & x > 0 \\ 1 & x = 0 \\ -2x + 3 & x < 0 \end{cases}$
  6. Draw a curve in the plane that is not the graph of a function.
  7. Is this expression a function? Why or why not? \[f(x) = \begin{cases}x^2 - 3 & x \geq 0 \\ -x - 4 & x \leq 0\end{cases}\]
  8. Given the two functions $f$ and $g$
    1. What is $f(0)$?
    2. What is $g(3)$?
    3. Solve the equation $f(x) = g(x)$.
    4. Solve the equation $f(x) < g(x)$.
    5. Solve the equation $f(x) \geq g(x)$.
    6. On what intervals is $f(x)$ increasing and decreasing?
    7. On what intervals is $g(x)$ increasing and decreasing?
    8. What is the local maxima of $f(x)$?
  9. Solve the following equations/inequalities with 1.5 methods or using the graph. You can use Desmos (but graph both graphs on the homework you will submit).
    1. $x - 2 = 4 - x$
    2. $x^3 + 3x^2 = -x^2 + 5x$
    3. $x^3 + 3x^2 < -x^2 + 5x$
    4. $16x^3 + 16x^2 = x + 1$
    5. $1 + \sqrt{x} = \sqrt{x^2 + 1}$
  10. Find the increasing/decreasing intervals and all local maxima/minima and the location at which they occur:
  11. What is the average rate of change of $f(x)$ on $(x, x + h)$ defined as?
  12. If a function has negative average rate of change on $(a, b)$, must it be decreasing on $(a, b)$? If not, draw a counterexample.
  13. Find the average rate of change for the following functions between the given values. Fully simplify when necessary.
    1. $f(x) = \sqrt{x}; \qquad (x, x + h)$
      Hint Rationalize the numerator.
    2. $f(x) = x^3 - 4x^2; \qquad (0, 10)$
    3. $f(x) = 3x - 2; \qquad (2, 3)$
    4. $f(x) = 1 - 3x^2; \qquad (2, 2 + h)$
    5. $f(x) = \dfrac{1}{x}; \qquad (x, x + h)$
  14. Which transformations should be last?
  15. If I have a base function $f(x)$, explain in English what the following transformations will do to $f(x)$:
    1. $f(x + 2)$
    2. $f(-x)$
    3. $-f(x) - 3$
    4. $-f(-2x + 4)$
    5. $3-f\left(-\frac{1}{2}x + 2\right)$
  16. Suppose \[f(x) = \sqrt{x} \qquad g(x) = \sqrt{-x - 2} \qquad h(x) = \sqrt{- x + 2} \qquad i(x) = \sqrt{-(x + 2)}\]
    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. Suppose \[f(x) = x^3 \qquad g(x) = \left(-x - 4\right)^3 \qquad h(x) = \left(-x + 4\right)^3 \qquad i(x) = (-(x+4))^3\]
    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?
  18. Write out the list of transformations in the order needed to transform the parent function into $f(x)$.
    1. $f(x) = \sqrt{x + 4} - 3$
      Hint Parent is $g(x) = \sqrt{x}$
    2. $f(x) = 2-(x-1)^2$
      Hint Parent is $g(x) = x^2$
    3. $f(x) = 4 - \sqrt{-2x + 4}$
      Hint Parent is $g(x) = \sqrt{x}$. Don't forget to factor out $-2$.
    4. $f(x) = -2\lvert{-x - 1}\rvert + 3$
      Hint Parent is $g(x) = \lvert x \rvert$. Don't forget to factor out $-1$.