{"id":1013,"date":"2025-07-21T19:16:00","date_gmt":"2025-07-21T19:16:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1013"},"modified":"2026-01-14T18:36:45","modified_gmt":"2026-01-14T18:36:45","slug":"rational-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-learn-it-2\/","title":{"raw":"Rational Functions: Learn It 3","rendered":"Rational Functions: Learn It 3"},"content":{"raw":"

Domain and Its Connection to Vertical Asymptotes<\/h2>\r\nA vertical asymptote<\/strong> represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function (a special case of a rational function) cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.\r\n\r\n
\r\n

domain of a rational function<\/h3>\r\nThe domain of a rational function includes all real numbers except those that cause the denominator to equal zero.\r\n\r\n<\/section>
How To: Given a rational function, find the domain.<\/strong>\r\n
    \r\n \t
  1. Set the denominator equal to zero.<\/li>\r\n \t
  2. Solve to find the values of the variable that cause the denominator to equal zero.<\/li>\r\n \t
  3. The domain contains all real numbers except those found in Step 2.<\/li>\r\n<\/ol>\r\n<\/section>
    Find the domain of [latex]f\\left(x\\right)=\\dfrac{x+3}{{x}^{2}-9}[\/latex].[reveal-answer q=\"860212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"860212\"]Begin by setting the denominator equal to zero and solving.\r\n

    [latex]\\begin{align} {x}^{2}-9&=0 \\\\ {x}^{2}&=9 \\\\ x&=\\pm 3 \\end{align}[\/latex]<\/p>\r\nThe denominator is equal to zero when [latex]x=\\pm 3[\/latex]. The domain of the function is all real numbers except [latex]x=\\pm 3[\/latex].\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nA graph of this function confirms that the function is not defined when [latex]x=\\pm 3[\/latex].\r\n\r\n\"Graph\r\n\r\nThere is a vertical asymptote at [latex]x=3[\/latex] and a hole in the graph at [latex]x=-3[\/latex]. We will discuss these types of holes in greater detail later in this section.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]318942[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]318943[\/ohm_question]<\/section>By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes. We may even be able to approximate their location. Even without the graph, however, we can still determine whether a given rational function has any asymptotes, and calculate their location.\r\n

    Vertical Asymptotes<\/h3>\r\nThe vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.\r\n\r\n
    How To: Given a rational function, identify any vertical asymptotes of its graph.<\/strong>\r\n
      \r\n \t
    1. Factor the numerator and denominator.<\/li>\r\n \t
    2. Note any restrictions in the domain of the function.<\/li>\r\n \t
    3. Reduce the expression by canceling common factors in the numerator and the denominator.<\/li>\r\n \t
    4. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical asymptotes occur.<\/li>\r\n \t
    5. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities.<\/li>\r\n<\/ol>\r\n<\/section>
      Find the vertical asymptote and the domain of [latex]f\\left(x\\right)=\\dfrac{x+3}{{x}^{2}-9}[\/latex].[reveal-answer q=\"8602123\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"8602123\"]\r\n
        \r\n \t
      • Identify the denominator and set it equal to zero.<\/strong><\/li>\r\n<\/ul>\r\n

        [latex]\\begin{align} {x}^{2}-9&=0 \\\\ {x}^{2}&=9 \\\\ x&=\\pm 3 \\end{align}[\/latex]<\/p>\r\nThe denominator is equal to zero when [latex]x=\\pm 3[\/latex].\r\n\r\nThus, the domain of the function is all real numbers except [latex]x=\\pm 3[\/latex]<\/strong> or in interval notation:\u00a0[latex](-\\infty, -3) \\cup (-3, 3) \\cup (3, \\infty)[\/latex].<\/strong>\r\n

          \r\n \t
        • Check the numerator at these values:<\/strong>\u00a0If both the numerator and the denominator are zero at the same [latex]x[\/latex]-value, this indicates a hole in the graph rather than a vertical asymptote.<\/li>\r\n<\/ul>\r\nThe numerator [latex]x+3[\/latex] does not equal zero when [latex]x=3[\/latex]. However, it equals to zero when [latex]x=-3[\/latex]. So:\r\n
            \r\n \t
          • \u00a0[latex]x=3[\/latex] is a vertical asymptote.<\/li>\r\n \t
          • [latex]x = -3[\/latex] is a hole in the graph, not a vertical asymptote.<\/li>\r\n<\/ul>\r\nA graph of this function confirms our finding above.\r\n\r\n\"Graph\r\n\r\nVertical asymptote: [latex]x=3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n

            Removable Discontinuities<\/h3>\r\nOccasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity<\/strong>.\r\n\r\n
            \r\n

            removable discontinuities of rational functions<\/h3>\r\nA removable discontinuity<\/strong> occurs in the graph of a rational function at [latex]x=a[\/latex] if a<\/em>\u00a0is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find any, we set the common factor equal to 0 and solve. This is the location of the removable discontinuity.\r\n\r\n \r\n
              \r\n \t
            • This is true if the multiplicity of this factor is greater than or equal to that in the denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that value.<\/li>\r\n<\/ul>\r\n<\/section>
              For example, the function [latex]f\\left(x\\right)=\\dfrac{{x}^{2}-1}{{x}^{2}-2x - 3}[\/latex] may be re-written by factoring the numerator and the denominator.\r\n

              [latex]f\\left(x\\right)=\\dfrac{\\left(x+1\\right)\\left(x - 1\\right)}{\\left(x+1\\right)\\left(x - 3\\right)}[\/latex]<\/p>\r\nNotice that [latex]x+1[\/latex] is a common factor to the numerator and the denominator. The zero of this factor, [latex]x=-1[\/latex], is the location of the removable discontinuity. Notice also that [latex]x - 3[\/latex] is not a factor in both the numerator and denominator. The zero of this factor, [latex]x=3[\/latex], is the vertical asymptote.\r\n\r\n\"Graph\r\n\r\n<\/section>

              Find the vertical asymptotes and removable discontinuities of the graph of [latex]k\\left(x\\right)=\\dfrac{x - 2}{{x}^{2}-4}[\/latex].[reveal-answer q=\"519186\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"519186\"]Factor the numerator and the denominator.\r\n

              [latex]k\\left(x\\right)=\\dfrac{x - 2}{\\left(x - 2\\right)\\left(x+2\\right)}[\/latex]<\/p>\r\nNotice that there is a common factor in the numerator and the denominator, [latex]x - 2[\/latex]. The zero for this factor is [latex]x=2[\/latex]. This is the location of the removable discontinuity.\r\n\r\nNotice that there is a factor in the denominator that is not in the numerator, [latex]x+2[\/latex]. The zero for this factor is [latex]x=-2[\/latex]. The vertical asymptote is [latex]x=-2[\/latex].\r\n\r\n\"Graph\r\n\r\nThe graph of this function will have the vertical asymptote at [latex]x=-2[\/latex], but at [latex]x=2[\/latex] the graph will have a hole.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

              [ohm2_question hide_question_numbers=1]24652[\/ohm2_question]<\/section>
              [ohm_question hide_question_numbers=1]294374[\/ohm_question]<\/section>","rendered":"

              Domain and Its Connection to Vertical Asymptotes<\/h2>\n

              A vertical asymptote<\/strong> represents a value at which a rational function is undefined, so that value is not in the domain of the function. A reciprocal function (a special case of a rational function) cannot have values in its domain that cause the denominator to equal zero. In general, to find the domain of a rational function, we need to determine which inputs would cause division by zero.<\/p>\n

              \n

              domain of a rational function<\/h3>\n

              The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.<\/p>\n<\/section>\n

              How To: Given a rational function, find the domain.<\/strong><\/p>\n
                \n
              1. Set the denominator equal to zero.<\/li>\n
              2. Solve to find the values of the variable that cause the denominator to equal zero.<\/li>\n
              3. The domain contains all real numbers except those found in Step 2.<\/li>\n<\/ol>\n<\/section>\n
                Find the domain of [latex]f\\left(x\\right)=\\dfrac{x+3}{{x}^{2}-9}[\/latex].<\/p>\n