Homework 6

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. For each red function, write out what the transformation from $f(x)$ should be:
  2. Graph the following functions using transformations. Include at least two transformed points and ordered of transformations you apply them in for full credit.
    1. $g(x) = \sqrt{x + 4} - 3$
    2. $g(x) = -(x-1)^2$
    3. $g(x) = -2 - \sqrt{-2x + 6}$
    4. $g(x) = -\lvert-2x - 2\rvert + 4$
    5. $g(x) = 2(-2x+6)^2 - 1$
  3. Suppose \[f(x) = 2x^2 + x \qquad g(x) = x^2 - x \qquad h(x) = x^2 - 1 \qquad k(x) = x^4\] Evaluate and expand the following:
    1. $f(x) + g(x) + h(x)$
    2. $f(x) - g(x)$
      Hint Look at how many terms you are substituting. Don't forget...
    3. $g(x)h(x) - k(x)$
    4. $f(x)\left[k(x)\right]^2$
    5. $\dfrac{g(x)}{h(x)}$ (also find the domain).
  4. Consider $f(x) = x^2 -1$ and $g(x) = x^2 + 2x + 1$. We make a new function $h(x) = \dfrac{g(x)}{f(x)}$.
    1. Find and simplify the formula for $h(x)$.
    2. What are the $x$-values of the holes in $h(x)$?
    3. What is the domain of $h(x)$?
  5. Suppose \begin{align}f(x) &= (x^2 - 1)^3 \\ f'(x) &= 6x(x^2 - 1)^2 \\ g(x) &= x^2 + 1 \\ g'(x) &= 2x\end{align} Evaluate and simplify the following:
    1. $\dfrac{g'(x)f(x) - f'(x)g(x)}{\left[g(x)\right]^2}$
      Hint Look at how many terms you are substituting. Don't forget...
    2. $f'(x)g(x) + f(x)g'(x)$
  6. Given the functions $f(x) = 2x- 5$ and $g(x) = 3 - x^2$, evaluate the following and fully simplify.
    Hint Expand because like terms are created.
    1. $f\circ g$
    2. $f(g(0))$
    3. $g \circ f$
    4. $f \circ f$
    5. $g \circ g$
    6. $f \circ f \circ f$
  7. Given the following functions $F(x)$, find two functions $f$ and $g$ where $f\circ g = F$.
    Choosing $f(x) = x$ or $g(x) = x$ is zero credit for this problem.
    1. $F(x) = (x-3)^4$
    2. $F(x) = (2x-3)^{-\frac{1}{2}}$
    3. $F(x) = \sqrt{1 - \sqrt{x}}$
    4. $F(x) = \sqrt{(x^2 + 2x + 3)^3}$
  8. 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?
  9. Suppose $f$ is not a one-to-one function. Explain what property is violated when we try to define $f^{-1}$.
  10. 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.
  11. 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?
  12. Find the codomain of the following functions.
    1. $f(x) = \sqrt{x}$
    2. $f(x) = x^2 - 77$
      Hint Transformations.
    3. $f(x) = \lvert x \rvert + 1$
  13. 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. $f(x) = \dfrac{2x + 4}{x - 7}$
  14. 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 $y$?
  15. leave blank, accidentally copied previous problem.
  16. 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. $f(x) = -3x^2 + 6x$
  17. 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. $g(x) = 2x^2 + 8x + 11$