{"id":1418,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1418"},"modified":"2015-11-12T18:35:29","modified_gmt":"2015-11-12T18:35:29","slug":"find-the-domains-of-rational-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/find-the-domains-of-rational-functions\/","title":{"raw":"Find the domains of rational functions","rendered":"Find the domains of rational functions"},"content":{"raw":"

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 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

\n

A General Note: 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\n<\/div>\n

\n

How To: Given a rational function, find the domain.<\/h3>\n
  1. Set the denominator equal to zero.<\/li>\n\t
  2. Solve to find the x<\/em>-values that cause the denominator to equal zero.<\/li>\n\t
  3. The domain is all real numbers except those found in Step 2.<\/li>\n<\/ol><\/div>\n
    \n
    \n
    \n

    Example 4: Finding the Domain of a Rational Function<\/h3>\n

    Find the domain of [latex]f\\left(x\\right)=\\frac{x+3}{{x}^{2}-9}\\\\[\/latex].<\/p>\n\n<\/div>\n

    \n

    Solution<\/h3>\n

    Begin by setting the denominator equal to zero and solving.<\/p>\n\n

    [latex]\\begin{cases} {x}^{2}-9=0 \\hfill \\\\ \\text{ }{x}^{2}=9\\hfill \\\\ \\text{ }x=\\pm 3\\hfill \\end{cases}\\\\[\/latex]<\/div>\n

    The 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].<\/p>\n\n<\/div>\n

    \n

    Analysis of the Solution<\/h3>\n

    A graph of this function confirms that the function is not defined when [latex]x=\\pm 3\\\\[\/latex].<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 8<\/b>[\/caption]\n

    There 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.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n

    \n

    Try It 4<\/h3>\n

    Find the domain of [latex]f\\left(x\\right)=\\frac{4x}{5\\left(x - 1\\right)\\left(x - 5\\right)}\\\\[\/latex].<\/p>\nSolution<\/a>\n\n<\/div>","rendered":"

    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 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

    A General Note: 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<\/div>\n

    \n

    How To: Given a rational function, find the domain.<\/h3>\n
      \n
    1. Set the denominator equal to zero.<\/li>\n
    2. Solve to find the x<\/em>-values that cause the denominator to equal zero.<\/li>\n
    3. The domain is all real numbers except those found in Step 2.<\/li>\n<\/ol>\n<\/div>\n
      \n
      \n
      \n

      Example 4: Finding the Domain of a Rational Function<\/h3>\n

      Find the domain of [latex]f\\left(x\\right)=\\frac{x+3}{{x}^{2}-9}\\\\[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      Begin by setting the denominator equal to zero and solving.<\/p>\n

      [latex]\\begin{cases} {x}^{2}-9=0 \\hfill \\\\ \\text{ }{x}^{2}=9\\hfill \\\\ \\text{ }x=\\pm 3\\hfill \\end{cases}\\\\[\/latex]<\/div>\n

      The 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].<\/p>\n<\/div>\n

      \n

      Analysis of the Solution<\/h3>\n

      A graph of this function confirms that the function is not defined when [latex]x=\\pm 3\\\\[\/latex].<\/p>\n

      \"Graph<\/p>\n

      Figure 8<\/b><\/p>\n<\/div>\n

      There 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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Try It 4<\/h3>\n

      Find the domain of [latex]f\\left(x\\right)=\\frac{4x}{5\\left(x - 1\\right)\\left(x - 5\\right)}\\\\[\/latex].<\/p>\n

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

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