{"id":11388,"date":"2015-07-16T22:05:49","date_gmt":"2015-07-16T22:05:49","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11388"},"modified":"2018-07-07T20:10:07","modified_gmt":"2018-07-07T20:10:07","slug":"find-domains-and-ranges-of-the-toolkit-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/find-domains-and-ranges-of-the-toolkit-functions\/","title":{"raw":"Find domains and ranges of the toolkit functions","rendered":"Find domains and ranges of the toolkit functions"},"content":{"raw":"
\r\n\r\nWe will now return to our set of toolkit functions to determine the domain and range of each.\r\n\r\n<\/div>\r\n
\r\n
\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Constant 11[\/caption]\r\n

Figure 11.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Identity 12[\/caption]\r\n

Figure 12.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Absolute 13[\/caption]\r\n

Figure 13.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Quadratic 14[\/caption]\r\n

Figure 14.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Cubic 15[\/caption]\r\n

Figure 15.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Reciprocal 16[\/caption]\r\n

Figure 16.<\/strong> For the reciprocal function [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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Reciprocal 17[\/caption]\r\n

Figure 17.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Square 18[\/caption]\r\n

Figure 18.<\/strong> 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>\r\n<\/figure>\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Cube 19[\/caption]\r\n

Figure 19.<\/strong> 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>\r\n<\/figure>\r\n

\r\n

How To: Given the formula for a function, determine the domain and range.<\/h3>\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<\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 8: Finding the Domain and Range Using Toolkit Functions<\/h3>\r\n

    Find the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\n

    There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\r\n

    The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] and the range is also [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n
    \r\n

    Example 9: Finding the Domain and Range<\/h3>\r\n

    Find the domain and range of [latex]f\\left(x\\right)=\\frac{2}{x+1}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\n

    We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right)[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Example 10: Finding the Domain and Range<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n

    We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\r\n\r\n

    [latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/div>\r\n

    The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].<\/p>\r\n

    We then find the range. We know that [latex]f\\left(-4\\right)=0[\/latex], and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of [latex]f[\/latex] is [latex]\\left[0,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    Analysis of the Solution<\/h3>\r\nFigure 20\u00a0represents the function [latex]f[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 20<\/b>[\/caption]\r\n\r\n
    \r\n

    Try It 7<\/h3>\r\nFind the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>","rendered":"
    \n

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

    \n
    \n
    \"Constant<\/p>\n

    11<\/p>\n<\/div>\n

    Figure 11.<\/strong> 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<\/figure>\n

    \n
    \"Identity<\/p>\n

    12<\/p>\n<\/div>\n

    Figure 12.<\/strong> 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<\/figure>\n

    \n
    \"Absolute<\/p>\n

    13<\/p>\n<\/div>\n

    Figure 13.<\/strong> 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<\/figure>\n

    \n
    \"Quadratic<\/p>\n

    14<\/p>\n<\/div>\n

    Figure 14.<\/strong> 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<\/figure>\n

    \n
    \"Cubic<\/p>\n

    15<\/p>\n<\/div>\n

    Figure 15.<\/strong> 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<\/figure>\n

    \n
    \"Reciprocal<\/p>\n

    16<\/p>\n<\/div>\n

    Figure 16.<\/strong> For the reciprocal function [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<\/figure>\n

    \n
    \"Reciprocal<\/p>\n

    17<\/p>\n<\/div>\n

    Figure 17.<\/strong> 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<\/figure>\n

    \n
    \"Square<\/p>\n

    18<\/p>\n<\/div>\n

    Figure 18.<\/strong> 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<\/figure>\n

    \n
    \"Cube<\/p>\n

    19<\/p>\n<\/div>\n

    Figure 19.<\/strong> 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<\/figure>\n

    \n

    How To: Given the formula for a function, determine the domain and range.<\/h3>\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<\/div>\n
      \n
      \n
      \n

      Example 8: Finding the Domain and Range Using Toolkit Functions<\/h3>\n

      Find the domain and range of [latex]f\\left(x\\right)=2{x}^{3}-x[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\n

      The domain is [latex]\\left(-\\infty ,\\infty \\right)[\/latex] and the range is also [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n
      \n

      Example 9: Finding the Domain and Range<\/h3>\n

      Find the domain and range of [latex]f\\left(x\\right)=\\frac{2}{x+1}[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      We cannot evaluate the function at [latex]-1[\/latex] because division by zero is undefined. The domain is [latex]\\left(-\\infty ,-1\\right)\\cup \\left(-1,\\infty \\right)[\/latex]. Because the function is never zero, we exclude 0 from the range. The range is [latex]\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Example 10: Finding the Domain and Range<\/h3>\n

      Find the domain and range of [latex]f\\left(x\\right)=2\\sqrt{x+4}[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n

      [latex]x+4\\ge 0\\text{ when }x\\ge -4[\/latex]<\/div>\n

      The domain of [latex]f\\left(x\\right)[\/latex] is [latex]\\left[-4,\\infty \\right)[\/latex].<\/p>\n

      We then find the range. We know that [latex]f\\left(-4\\right)=0[\/latex], and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of [latex]f[\/latex] is [latex]\\left[0,\\infty \\right)[\/latex].<\/p>\n<\/div>\n

      Analysis of the Solution<\/h3>\n

      Figure 20\u00a0represents the function [latex]f[\/latex].<\/p>\n

      \"Graph<\/p>\n

      Figure 20<\/b><\/p>\n<\/div>\n

      \n

      Try It 7<\/h3>\n

      Find the domain and range of [latex]f\\left(x\\right)=-\\sqrt{2-x}[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n<\/section>\n\n\t\t\t

      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
      \n\t\t\t\t\t\t\t
      CC licensed content, Shared previously<\/div>