{"id":845,"date":"2024-04-08T17:02:51","date_gmt":"2024-04-08T17:02:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=845"},"modified":"2024-08-05T12:11:51","modified_gmt":"2024-08-05T12:11:51","slug":"review-of-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/review-of-functions-learn-it-3\/","title":{"raw":"Review of Functions: Learn It 3","rendered":"Review of Functions: Learn It 3"},"content":{"raw":"

Intercepts of a Function<\/h2>\r\n

Intercepts are a key feature when graphing and analyzing functions because they provide critical points at which the graph intersects the axes.<\/p>\r\n

The points where the graph of the function intersects the [latex] x[\/latex]-axis are known as the [latex]x[\/latex]-intercept. The [latex]x[\/latex]-intercept indicates where the output [latex]f(x)[\/latex] is [latex]0[\/latex].<\/p>\r\n

These intercepts are also known as the zeros<\/strong> or roots<\/strong> of the function because they satisfy the equation [latex]f(x)=0[\/latex]. The [latex]x[\/latex]-intercepts determine where the graph of [latex]f[\/latex] intersects the [latex]x[\/latex]-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the [latex]x[\/latex]-axis, or it may intersect multiple (or even infinitely many) times.<\/p>\r\n

Another point of interest is the [latex]y[\/latex]-intercept, if it exists. The [latex]y[\/latex]-intercept<\/strong> of a function is the point where the graph of the function crosses the [latex]y[\/latex]-axis. It represents the output value when the input value [latex]x[\/latex] is [latex]0[\/latex]. In other words, it\u2019s the value of the function [latex]f(x)[\/latex] at [latex]x=0[\/latex], given by [latex](0,f(0))[\/latex].<\/p>\r\n

\r\n

How to: Given a Function [latex]f\\left(x\\right)[\/latex], Find the [latex]y[\/latex]- and [latex]x[\/latex]-intercepts<\/strong><\/p>\r\n

Finding the [latex]y[\/latex]-intercept:<\/strong><\/p>\r\n

    \r\n\t
  1. Plug in zero for the [latex]x[\/latex]-value in the function and solve for [latex]f(0)[\/latex].<\/li>\r\n\t
  2. The y-intercept will be at the point [latex](0,f(0))[\/latex].<\/li>\r\n<\/ol>\r\n

    Finding the [latex]x[\/latex]-intercept:<\/strong><\/p>\r\n

      \r\n\t
    1. Set the function equal to zero,[latex]f(x)=0[\/latex], and solve for [latex]x[\/latex]\u00a0to find the roots of the function.<\/li>\r\n\t
    2. The solutions are the [latex]x[\/latex]-intercepts, and they'll be in the form [latex](x,0)[\/latex], where [latex]x[\/latex]\u00a0represents each root.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
      \r\n

      Since a function has exactly one output for each input, the graph of a function can have, at most, one [latex]y[\/latex]-intercept. If [latex]x=0[\/latex] is in the domain of a function [latex]f[\/latex], then [latex]f[\/latex] has exactly one [latex]y[\/latex]-intercept. If [latex]x=0[\/latex] is not in the domain of [latex]f[\/latex], then [latex]f[\/latex] has no [latex]y[\/latex]-intercept.<\/p>\r\n<\/section>\r\n

      \r\n

      Consider the function [latex]f(x)=-4x+2[\/latex].<\/p>\r\n

        \r\n\t
      1. Find all zeros of [latex]f[\/latex].<\/li>\r\n\t
      2. Find the [latex]y[\/latex]-intercept (if any).<\/li>\r\n<\/ol>\r\n

        [reveal-answer q=\"fs-id1170572111812\"]Show Solution[\/reveal-answer]
        \r\n[hidden-answer a=\"fs-id1170572111812\"]<\/p>\r\n

          \r\n\t
        1. To find the zeros, solve [latex]f(x)=-4x+2=0[\/latex]. We discover that [latex]f[\/latex] has one zero at [latex]x=\\frac{1}{2}[\/latex].<\/li>\r\n\t
        2. The [latex]y[\/latex]-intercept is given by [latex](0,f(0))=(0,2)[\/latex].<\/li>\r\n<\/ol>\r\n

          [\/hidden-answer]<\/p>\r\n<\/section>\r\n

          \r\n

          [ohm_question hide_question_numbers=1]218410[\/ohm_question]<\/p>\r\n

          [ohm_question hide_question_numbers=1]287051[\/ohm_question]<\/p>\r\n

          [ohm_question hide_question_numbers=1]287050[\/ohm_question]<\/p>\r\n<\/section>","rendered":"

          Intercepts of a Function<\/h2>\n

          Intercepts are a key feature when graphing and analyzing functions because they provide critical points at which the graph intersects the axes.<\/p>\n

          The points where the graph of the function intersects the [latex]x[\/latex]-axis are known as the [latex]x[\/latex]-intercept. The [latex]x[\/latex]-intercept indicates where the output [latex]f(x)[\/latex] is [latex]0[\/latex].<\/p>\n

          These intercepts are also known as the zeros<\/strong> or roots<\/strong> of the function because they satisfy the equation [latex]f(x)=0[\/latex]. The [latex]x[\/latex]-intercepts determine where the graph of [latex]f[\/latex] intersects the [latex]x[\/latex]-axis, which gives us more information about the shape of the graph of the function. The graph of a function may never intersect the [latex]x[\/latex]-axis, or it may intersect multiple (or even infinitely many) times.<\/p>\n

          Another point of interest is the [latex]y[\/latex]-intercept, if it exists. The [latex]y[\/latex]-intercept<\/strong> of a function is the point where the graph of the function crosses the [latex]y[\/latex]-axis. It represents the output value when the input value [latex]x[\/latex] is [latex]0[\/latex]. In other words, it\u2019s the value of the function [latex]f(x)[\/latex] at [latex]x=0[\/latex], given by [latex](0,f(0))[\/latex].<\/p>\n

          \n

          How to: Given a Function [latex]f\\left(x\\right)[\/latex], Find the [latex]y[\/latex]– and [latex]x[\/latex]-intercepts<\/strong><\/p>\n

          Finding the [latex]y[\/latex]-intercept:<\/strong><\/p>\n

            \n
          1. Plug in zero for the [latex]x[\/latex]-value in the function and solve for [latex]f(0)[\/latex].<\/li>\n
          2. The y-intercept will be at the point [latex](0,f(0))[\/latex].<\/li>\n<\/ol>\n

            Finding the [latex]x[\/latex]-intercept:<\/strong><\/p>\n

              \n
            1. Set the function equal to zero,[latex]f(x)=0[\/latex], and solve for [latex]x[\/latex]\u00a0to find the roots of the function.<\/li>\n
            2. The solutions are the [latex]x[\/latex]-intercepts, and they’ll be in the form [latex](x,0)[\/latex], where [latex]x[\/latex]\u00a0represents each root.<\/li>\n<\/ol>\n<\/section>\n
              \n

              Since a function has exactly one output for each input, the graph of a function can have, at most, one [latex]y[\/latex]-intercept. If [latex]x=0[\/latex] is in the domain of a function [latex]f[\/latex], then [latex]f[\/latex] has exactly one [latex]y[\/latex]-intercept. If [latex]x=0[\/latex] is not in the domain of [latex]f[\/latex], then [latex]f[\/latex] has no [latex]y[\/latex]-intercept.<\/p>\n<\/section>\n

              \n

              Consider the function [latex]f(x)=-4x+2[\/latex].<\/p>\n

                \n
              1. Find all zeros of [latex]f[\/latex].<\/li>\n
              2. Find the [latex]y[\/latex]-intercept (if any).<\/li>\n<\/ol>\n