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4.11: Piecewise-Definition Functions

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    Definition: Piecewise-Defined Functions

    Piecewise-Defined Functions are functions that are defined using different equations for different parts of the domain.

    Example 4.11.1

    Evaluate the following piecewise-defined function for the given values of \(x\), and graph the function:

    \(f(x) = \left\{\begin{array}{cc}−2x + 1 & −1 \leq x < 0 \\ x^2 + 2 &0 \leq x \leq 2\end{array} \right.\)

    Solution

    To graph this function, make a table of solutions:

    Table of Solutions for \(f(x) = −2x + 1 \)

    Domain \(−1 \leq x < 0\)
    \(x\) \(f(x)\)
    -1 3
    0 1 (open circle here, 0 not in the domain)

    Table of Solutions for \(f(x) = x^2 + 2\)

    Domain \(0 \leq x \leq 2\)
    \(x\) \(f(x)\)
    0 2
    1 3
    2 6
    clipboard_e94fd5197718a7373772af1280306cf06.png
    Figure 4.11.1
    Example 4.11.2

    Evaluate the following piecewise-defined function for the given values of \(x\), and graph the function:

    \(f(x) = \left\{\begin{array}{cc} −x + 1 &x \leq −1 \\ 2 & −1 < x \leq 1 \\ −x + 3 &x > 1 \end{array}\right.\)

    Solution

    To graph this function, once again make a table of solutions:

    Table of Solutions for \(f(x) = −x + 1\)

    Domain \(x \leq −1\)
    \(x\) \(f(x)\)
    -3 4
    -2 3
    -1 2 (closed circle here, -1 is in the domain)

    Table of Solutions for \(f(x) = 2\)

    Domain \(−1 < x \leq 1\)
    \(x\) \(f(x)\)
    -1 2 (open circle filled in by the previous function, -1 not in the domain)
    0 2
    1 2 (closed circle here, 1 is in the domain)

    Table of Solutions for \(f(x) = −x + 3\)

    Domain \(x > 1\)
    \(x\) \(f(x)\)
    1 2 (open circle filled in by the previous function, 1 not in the domain)
    2 1
    3 0
    clipboard_e795cd2fce50083772c8741bdcad72855.png
    Figure 4.11.2
    Exercise 4.11.1

    Evaluate the following piecewise-defined functions for the given values of x, and graph the functions:.

    1. \(f(x)=\left\{\begin{array}{cc}
      x & x<0\\
      2 x+1 &x\geq 0
      \end{array}\right.\)
    2. \(g(x) = \left\{\begin{array}{cc} 4 − x& x < 2\\ 2x − 2 &x \geq 2 \end{array} \right.\)
    3. \(h(x) = \left\{\begin{array}{cc} −x − 1 & x < −1 \\ 0& −1 \leq x \leq 1 \\ x + 1 & x > 1 \end{array} \right.\)
    4. \(g(x) = \left\{\begin{array}{cc} 6 & −8 \leq x < −4 \\ 3 &−4 \leq x \leq 5 \end{array}\right.\)
    5. \(f(x) = \left\{\begin{array}{cc} −x + 1 & −1 \leq x < 1 \\ \sqrt{x − 1 } &1 \leq x \leq 5\end{array}\right.\)

    This page titled 4.11: Piecewise-Definition Functions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, & Sanaa Saykali (ASCCC Open Educational Resources Initiative) .