{"id":1472,"date":"2024-05-24T02:51:35","date_gmt":"2024-05-24T02:51:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1472"},"modified":"2025-11-20T15:23:02","modified_gmt":"2025-11-20T15:23:02","slug":"graphs-and-characteristics-of-basic-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphs-and-characteristics-of-basic-functions-learn-it-4\/","title":{"raw":"Domain and Range: Learn It 4","rendered":"Domain and Range: Learn It 4"},"content":{"raw":"

Domain and Range of Toolkit Functions<\/h2>\r\nWe will now return to our set of toolkit functions to determine the domain and range of each.\r\n\r\nFor the constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Constant Graph of constant function[\/caption]\r\n\r\nFor the identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Identity Graph of the identity function[\/caption]\r\n\r\nFor the absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Absolute Graph of absolute value function[\/caption]\r\n\r\nFor the quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Quadratic Graph of quadratic function[\/caption]\r\n\r\nFor the cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Cubic Graph of cubic function[\/caption]\r\n\r\nFor the reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Reciprocal Graph of reciprocal function[\/caption]\r\n\r\nFor the reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Reciprocal Graph of reciprocal squared function[\/caption]\r\n\r\nFor the square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Square Graph of square root function[\/caption]\r\n\r\nFor the cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Cube Graph of cube root function[\/caption]\r\n\r\n
\r\n

Given the formula for a function, determine the domain and range.<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. Exclude from the domain any input values that result in division by zero.<\/li>\r\n \t
  2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\r\n \t
  3. Use the valid input values to determine the range of the output values.<\/li>\r\n \t
  4. Look at the function graph and table values to confirm the actual function behavior.<\/li>\r\n<\/ol>\r\n<\/section>
    When evaluating functions to plot points, remember to wrap the variable in parentheses to be sure you handle negative input appropriately.Ex. For the function [latex]f(x) = x^2[\/latex], evaluate [latex]f(-3)[\/latex].[reveal-answer q=\"257996\"]more[\/reveal-answer]\r\n[hidden-answer a=\"257996\"][latex]\\begin{align}f(-3) &= (-3)^2 \\\\ &= 9\\end{align}[\/latex].[\/hidden-answer]<\/section>
    Find the domain and range of
    [latex]f(x) = \\dfrac{2}{x+1}[\/latex]<\/center>[reveal-answer q=\"746911\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"746911\"]We cannot evaluate the function at [latex]x = -1[\/latex] because division by zero is undefined. The domain is [latex](-\\infty, -1) \\cup (-1, \\infty)[\/latex]. Because the function is never zero, we exclude [latex]0[\/latex] from the range. The range is [latex](-\\infty, 0) \\cup (0, \\infty)[\/latex].[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]293806[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]293807[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]293808[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]292979[\/ohm_question]<\/section>","rendered":"

    Domain and Range of Toolkit Functions<\/h2>\n

    We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n

    For the constant function<\/strong> [latex]f\\left(x\\right)=c[\/latex], the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex]c[\/latex], so the range is the set [latex]\\left\\{c\\right\\}[\/latex] that contains this single element. In interval notation, this is written as [latex]\\left[c,c\\right][\/latex], the interval that both begins and ends with [latex]c[\/latex].<\/p>\n

    \"Constant
    Graph of constant function<\/figcaption><\/figure>\n

    For the identity function<\/strong> [latex]f\\left(x\\right)=x[\/latex], there is no restriction on [latex]x[\/latex]. Both the domain and range are the set of all real numbers.<\/p>\n

    \"Identity
    Graph of the identity function<\/figcaption><\/figure>\n

    For the absolute value function<\/strong> [latex]f\\left(x\\right)=|x|[\/latex], there is no restriction on [latex]x[\/latex]. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.<\/p>\n

    \"Absolute
    Graph of absolute value function<\/figcaption><\/figure>\n

    For the quadratic function<\/strong> [latex]f\\left(x\\right)={x}^{2}[\/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.<\/p>\n

    \"Quadratic
    Graph of quadratic function<\/figcaption><\/figure>\n

    For the cubic function<\/strong> [latex]f\\left(x\\right)={x}^{3}[\/latex], the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.<\/p>\n

    \"Cubic
    Graph of cubic function<\/figcaption><\/figure>\n

    For the reciprocal function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex], we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.<\/p>\n

    \"Reciprocal
    Graph of reciprocal function<\/figcaption><\/figure>\n

    For the reciprocal squared function<\/strong> [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/p>\n

    \"Reciprocal
    Graph of reciprocal squared function<\/figcaption><\/figure>\n

    For the square root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[]{x}[\/latex], we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[\/latex] is defined to be positive, even though the square of the negative number [latex]-\\sqrt{x}[\/latex] also gives us [latex]x[\/latex].<\/p>\n

    \"Square
    Graph of square root function<\/figcaption><\/figure>\n

    For the cube root function<\/strong> [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex], the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).<\/p>\n

    \"Cube
    Graph of cube root function<\/figcaption><\/figure>\n
    \n

    Given the formula for a function, determine the domain and range.<\/strong><\/p>\n

      \n
    1. Exclude from the domain any input values that result in division by zero.<\/li>\n
    2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n
    3. Use the valid input values to determine the range of the output values.<\/li>\n
    4. Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/section>\n
      When evaluating functions to plot points, remember to wrap the variable in parentheses to be sure you handle negative input appropriately.Ex. For the function [latex]f(x) = x^2[\/latex], evaluate [latex]f(-3)[\/latex].<\/p>\n