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2.2: Domain and Range

  • Page ID
    203383
    • Roy Simpson, Cosumnes River College
    • OpenStax

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    Prerequisite Skills
    • Sets and Numbers
      • Inequalities and Inequality Notation
      • Intervals and Interval Notation
    • Factoring Techniques
      • Factoring Difference of Squares
    • Solving Inequalities
      • Solving Linear Inequalities
      • Solving Compound Inequalities
      • Solving Quadratic Inequalities (sign chart method)
    • Graphs
      • Closed- and Open-Circle Notations

    Finding the Domain of a Function Defined by an Equation

    MyOpenMath \( \PageIndex{ 3 } \): Finding the domain and range of a relation given as a set of ordered pairs

    MyOpenMath \( \PageIndex{ 9 } \): Finding the domain of a quadratic function

    MyOpenMath \( \PageIndex{ 11 } \): Finding the domain of a rational function

    MyOpenMath \( \PageIndex{ 15 } \): Finding the domain of a function with an even root

    Using Notations to Specify Domain and Range

    Definition: Set-Builder Notation

    The notation\[ \{ x \mid x \text{ satisfies some condition} \} \nonumber \]is called set-builder notation. The braces, \(\{ \, \}\), are read literally as "the set of." The vertical bar is read as, "such that." Hence, set-builder notation is read as "the set of \(x\) such that \(x\) satisfies some given condition."

    Example: Using different notations

    State the solution to the last example using interval, inequality, and set-builder notations.

    Finding Domain and Range from Graphs

    MyOpenMath \( \PageIndex{ 6 } \): Finding domain and range from a graph

    Finding Domains and Ranges of the Toolkit Functions

    Let's state the domains and ranges of the Toolkit functions.

    Graphing Piecewise-Defined Functions

    Definition: Piecewise Function

    A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea as follows:\[ f ( x ) = \begin{cases}
    \text{formula 1,} & \text{if } x \text{ is in domain 1} \\[6pt]
    \text{formula 2,} & \text{if } x \text{ is in domain 2} \\[6pt]
    \text{formula 3,} & \text{if } x \text{ is in domain 3} \\[6pt]
    \vdots & \vdots \\[6pt]
    \end{cases} \nonumber \]

    MyOpenMath \(\PageIndex{16}\): Evaluating piecewise functions

    Example: Writing a piecewise function

    A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, \( n \), to the cost, \( C \).

    MyOpenMath \( \PageIndex{ 18 } \): Graphing a piecewise function

    Definition: Absolute Value Function

    The absolute value function is defined to be \( f(x) = |x| \), where\[ |\blacksquare| = \begin{cases}
    \blacksquare, & \text{ if } \blacksquare \geq 0 \\[6pt]
    -\blacksquare, & \text{ if } \blacksquare < 0 \\[6pt]
    \end{cases} \nonumber \]


    This page titled 2.2: Domain and Range is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Roy Simpson, Cosumnes River College (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.