{"id":1408,"date":"2024-05-23T02:38:25","date_gmt":"2024-05-23T02:38:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1408"},"modified":"2025-08-13T15:59:43","modified_gmt":"2025-08-13T15:59:43","slug":"introduction-to-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-functions-learn-it-2\/","title":{"raw":"Introduction to Functions: Learn It 2","rendered":"Introduction to Functions: Learn It 2"},"content":{"raw":"

Verifying a Function Using the Vertical Line Test<\/h2>\r\nWe can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.\r\n\r\nThe most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f(x)[\/latex] when the function is named [latex]f[\/latex].\r\n\r\nThe graph of the function is the set of all points [latex](x,y)[\/latex] in the plane that satisfies the equation [latex]y=f(x)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the [latex]x[\/latex]-coordinate of each point is an input value and the [latex]y[\/latex]-coordinate of each point is the corresponding output value.\r\n\r\n
\r\n\r\n[caption id=\"attachment_9189\" align=\"aligncenter\" width=\"434\"]\"Graph Graph of a polynomial[\/caption]\r\n\r\n<\/center> \r\n\r\nFor example, the black dots on the graph above tell us that [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. However, the set of all points [latex](x,y)[\/latex] satisfying [latex]y=f(x)[\/latex] is a curve. The curve shown includes [latex](0,2)[\/latex] and [latex](6,1)[\/latex] because the curve passes through those points.\r\n\r\nThe vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.\r\n\r\n
\r\n

vertical line test<\/h3>\r\nThe vertical line test<\/strong> determines if a relation is a function by checking that no vertical line intersects the graph more than once.\r\n\r\n<\/section>
How to: Use the Vertical Line Test<\/strong>Given a graph, use the vertical line test to determine if the graph represents a function following these steps.\r\n
    \r\n \t
  1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\r\n \t
  2. If there is any such line, determine that the graph does not represent a function.<\/li>\r\n \t
  3. If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\r\n<\/ol>\r\n<\/section>
    \r\n\r\n[caption id=\"attachment_9196\" align=\"aligncenter\" width=\"1024\"]\"Three Three graphs visually showing what is and is not a function[\/caption]\r\n\r\n<\/center>
    Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Graph Graphs of lines\/polynomials[\/caption]\r\n\r\n[reveal-answer q=\"689864\"]Show Solution[\/reveal-answer] [hidden-answer a=\"689864\"] If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of the graph above. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most [latex]x[\/latex]-values, a vertical line would intersect the graph at more than one point.
    \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a circle[\/caption]\r\n\r\n<\/center>[\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]13503[\/ohm2_question]<\/section>\r\n

    Determining Whether a Function is One-to-One<\/h2>\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown below, the stock price was [latex]$1000[\/latex] on five different dates, meaning that there were five different input values that all resulted in the same output value of [latex]$1000[\/latex].\r\n\r\n
    \r\n\r\n[caption id=\"attachment_9185\" align=\"aligncenter\" width=\"500\"]\"a A graph of market prices[\/caption]\r\n\r\n<\/center> \r\n\r\nHowever, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in the table below.\r\n\r\nThis grading system represents a one-to-one function<\/strong>, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.\r\n\r\n
    \r\n
    \r\n

    one-to-one function<\/h3>\r\nA one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<\/section>
    \r\n
      \r\n \t
    1. Is a balance a one-to-one function of the bank account number?<\/li>\r\n \t
    2. Is a bank account number a one-to-one function of the balance?<\/li>\r\n \t
    3. Is a balance a one-to-one function of the bank account number?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"997233\"]Show Solution[\/reveal-answer] [hidden-answer a=\"997233\"]\r\n
        \r\n \t
      1. Yes, because each bank account (input) has a single balance (output) at any given time.<\/span><\/li>\r\n \t
      2. No, because several bank accounts (inputs) may have the same balance (output).<\/span><\/li>\r\n \t
      3. No, because the more than one bank account (input) can have the same balance (output).<\/span><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]13502[\/ohm2_question]<\/section>\r\n

        The Horizontal Line Test<\/h2>\r\nOnce we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test<\/strong>. Draw horizontal lines through the graph. If we can draw any<\/em> horizontal line that intersects a graph more than once, then the graph does not<\/em> represent a one-to-one function because that [latex]y[\/latex] value has more than one input.\r\n\r\n
        \r\n

        horizontal line test<\/h3>\r\nThe horizontal line test<\/strong> checks if a function is one-to-one by ensuring that no horizontal line intersects the graph more than once.\r\n\r\n<\/section>
        How To: Horizontal Line Test<\/strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function following these steps.\r\n
          \r\n \t
        1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.<\/li>\r\n \t
        2. If there is any such line, the function is not one-to-one.<\/li>\r\n \t
        3. If no horizontal line can intersect the curve more than once, the function is one-to-one.<\/li>\r\n<\/ol>\r\n<\/section>
          Which of the graphs represent(s) a one-to-one function?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Graph Graphs of lines\/polynomials[\/caption]\r\n\r\n[reveal-answer q=\"173050\"]Show Solution[\/reveal-answer] [hidden-answer a=\"173050\"] The function in (a) is not one-to-one. The horizontal line shown below intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)
          \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Graph of a polynomial[\/caption]\r\n\r\n<\/center>The function in (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once. The function in (c) is not one-to-one. The horizontal line intersects the graph of the function at two points. [\/hidden-answer]<\/section>
          [ohm2_question hide_question_numbers=1]13504[\/ohm2_question]<\/section>","rendered":"

          Verifying a Function Using the Vertical Line Test<\/h2>\n

          We can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n

          The most common graphs name the input value [latex]x[\/latex] and the output value [latex]y[\/latex], and we say [latex]y[\/latex] is a function of [latex]x[\/latex], or [latex]y=f(x)[\/latex] when the function is named [latex]f[\/latex].<\/p>\n

          The graph of the function is the set of all points [latex](x,y)[\/latex] in the plane that satisfies the equation [latex]y=f(x)[\/latex]. If the function is defined for only a few input values, then the graph of the function is only a few points, where the [latex]x[\/latex]-coordinate of each point is an input value and the [latex]y[\/latex]-coordinate of each point is the corresponding output value.<\/p>\n

          \n
          \"Graph
          Graph of a polynomial<\/figcaption><\/figure>\n<\/div>\n

           <\/p>\n

          For example, the black dots on the graph above tell us that [latex]f(0)=2[\/latex] and [latex]f(6)=1[\/latex]. However, the set of all points [latex](x,y)[\/latex] satisfying [latex]y=f(x)[\/latex] is a curve. The curve shown includes [latex](0,2)[\/latex] and [latex](6,1)[\/latex] because the curve passes through those points.<\/p>\n

          The vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value.<\/p>\n

          \n

          vertical line test<\/h3>\n

          The vertical line test<\/strong> determines if a relation is a function by checking that no vertical line intersects the graph more than once.<\/p>\n<\/section>\n

          How to: Use the Vertical Line Test<\/strong>Given a graph, use the vertical line test to determine if the graph represents a function following these steps.<\/p>\n
            \n
          1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\n
          2. If there is any such line, determine that the graph does not represent a function.<\/li>\n
          3. If no vertical line can intersect the curve more than once, the graph does represent a function.<\/li>\n<\/ol>\n<\/section>\n
            \n
            \"Three
            Three graphs visually showing what is and is not a function<\/figcaption><\/figure>\n<\/div>\n
            Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<\/p>\n
            \"Graph
            Graphs of lines\/polynomials<\/figcaption><\/figure>\n