Piecewise-Defined Functions

  1. Find the following, given that $y =
    \begin{cases}
    \sqrt{4-x^2}, & x \le 0\\
    -3, & 0 \lt x \le 4\\
    2x-9, & x \gt 4
    \end{cases} $

    1. the domain   $\ans{\displaystyle{[-2,\infty)}}$

    2. the range   $\ans{\displaystyle{\{-3\} \cup (-1,\infty)}}$

    3. the graph of this function   $\ians{}$

  2. Find the following, given that $y =
    \begin{cases}
    \sqrt{25-x^2}, & x \le 0\\
    2x+1, & 0 \lt x \lt 3\\
    -4, & x \ge 3
    \end{cases} $

    1. the domain   $\ans{\displaystyle{[-5,\infty)}}$

    2. the range   $\ans{\displaystyle{\{-4\} \cup [0,7)}}$

    3. the graph of this function   $\ians{}$

  3. Graph each of the following piece-wise defined functions and indicate the domain and range:

    1. $\displaystyle{f(x) =
      \begin{cases}
      3, & x \le -3\\
      \left| x \right|, & -3 \lt x \le 3\\
      -3, & x \gt 3
      \end{cases}}$   $\ians{}$

    2. $\displaystyle{f(x) =
      \begin{cases}
      \sqrt{x+1}, & x \lt 3\\
      -2x+8, & x \ge 3
      \end{cases}}$   $\ians{}$

    3. $\displaystyle{f(x) =
      \begin{cases}
      \sqrt{4-x^2}, & x \le 0\\
      -\sqrt{x+4}, & 0 \lt x \le 5
      \end{cases}}$   $\ians{}$

    4. $\displaystyle{f(x) =
      \begin{cases}
      -x-1, & x \le -4\\
      3, & -4 \lt x \lt 2\\
      -x+5, & x \ge 2
      \end{cases}}$   $\ians{}$