{"id":1436,"date":"2024-05-23T04:46:55","date_gmt":"2024-05-23T04:46:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1436"},"modified":"2025-08-21T23:29:27","modified_gmt":"2025-08-21T23:29:27","slug":"introduction-to-functions-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-functions-learn-it-5\/","title":{"raw":"Introduction to Functions: Learn It 5","rendered":"Introduction to Functions: Learn It 5"},"content":{"raw":"

Finding Input and Output Values of a Function<\/h2>\r\n

Evaluating a Function Given in Tabular Form<\/h3>\r\nAs we have seen in this section, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables.\r\n\r\nFor example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of [latex]3[\/latex] seconds, but this is just a myth. Goldfish can remember up to [latex]3[\/latex] months, while the beta fish has a memory of up to [latex]5[\/latex] months. And while a puppy\u2019s memory span is no longer than [latex]30[\/latex] seconds, the adult dog can remember for [latex]5[\/latex] minutes. This is meager compared to a cat, whose memory span lasts for [latex]16[\/latex] hours.\r\n\r\nThe function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See the table below.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Pet<\/th>\r\nMemory span in hours<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Puppy<\/td>\r\n[latex]0.008[\/latex]<\/td>\r\n<\/tr>\r\n
Adult dog<\/td>\r\n[latex]0.083[\/latex]<\/td>\r\n<\/tr>\r\n
Cat<\/td>\r\n[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n
Goldfish<\/td>\r\n[latex]2160[\/latex]<\/td>\r\n<\/tr>\r\n
Beta fish<\/td>\r\n[latex]3600[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAt times, evaluating a function in table form may be more useful than using equations. Here let us call the function [latex]P[\/latex]. The domain<\/strong> of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \"goldfish.\" We would write [latex]P\\left(\\text{goldfish}\\right)=2160[\/latex]. Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.\r\n\r\n
How to: Given a function represented by a table, identify specific output and input values.<\/strong>\r\n
    \r\n \t
  1. Find the given input in the row (or column) of input values.<\/li>\r\n \t
  2. Identify the corresponding output value paired with that input value.<\/li>\r\n \t
  3. Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\r\n \t
  4. Identify the input value(s) corresponding to the given output value.<\/li>\r\n<\/ol>\r\n<\/section>
    Using the table below.\r\n
      \r\n \t
    1. Evaluate [latex]g\\left(3\\right)[\/latex].<\/li>\r\n \t
    2. Solve [latex]g\\left(n\\right)=6[\/latex].<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n\r\n
      [latex]n[\/latex]<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]g(n)[\/latex]<\/strong><\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"15206\"]Show Solution[\/reveal-answer] [hidden-answer a=\"15206\"]\r\n
        \r\n \t
      • Evaluating [latex]g\\left(3\\right)[\/latex] means determining the output value of the function [latex]g[\/latex] for the input value of [latex]n=3[\/latex]. The table output value corresponding to [latex]n=3[\/latex] is [latex]7[\/latex], so [latex]g\\left(3\\right)=7[\/latex].<\/li>\r\n \t
      • Solving [latex]g\\left(n\\right)=6[\/latex] means identifying the input values, [latex]n[\/latex], that produce an output value of [latex]6[\/latex]. The table below shows two solutions: [latex]n=2[\/latex] and [latex]n=4[\/latex].<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n
        [latex]n[\/latex]<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]g(n)[\/latex]<\/strong><\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nWhen we input [latex]2[\/latex] into the function [latex]g[\/latex], our output is [latex]6[\/latex]. When we input [latex]4[\/latex] into the function [latex]g[\/latex], our output is also [latex]6[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]13514[\/ohm2_question]<\/section>\r\n

        Finding Function Values from a Graph<\/h3>\r\nEvaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).\r\n\r\n
        We can view a function as a set of inputs and their corresponding outputs. That is, we can see a function as a set of ordered pairs, [latex]\\left(x, y \\right).[\/latex]Remember that, in function notation, [latex]y = f(x)[\/latex], so the ordered pairs containing inputs and outputs can be written in the form of (input<\/em>, output<\/em>) or [latex]\\left(x, f(x)\\right)[\/latex].<\/section>
        Given the graph below,\r\n
          \r\n \t
        1. Evaluate [latex]f\\left(2\\right)[\/latex].<\/li>\r\n \t
        2. Solve [latex]f\\left(x\\right)=4[\/latex].<\/li>\r\n<\/ol>\r\n \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a parabola[\/caption]\r\n\r\n[reveal-answer q=\"915833\"]Show Solution[\/reveal-answer] [hidden-answer a=\"915833\"]\r\n
            \r\n \t
          1. To evaluate [latex]f\\left(2\\right)[\/latex], locate the point on the curve where [latex]x=2[\/latex], then read the [latex]y[\/latex]-coordinate of that point. The point has coordinates [latex]\\left(2,1\\right)[\/latex], so [latex]f\\left(2\\right)=1[\/latex].
            \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a parabola with a plotted point[\/caption]\r\n\r\n \r\n\r\n<\/center><\/li>\r\n \t
          2. To solve [latex]f\\left(x\\right)=4[\/latex], we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line [latex]y=4[\/latex], we locate two points of the curve with output value [latex]4:[\/latex] [latex]\\left(-1,4\\right)[\/latex] and [latex]\\left(3,4\\right)[\/latex]. These points represent the two solutions to [latex]f\\left(x\\right)=4:[\/latex] [latex]x=-1[\/latex] or [latex]x=3[\/latex]. This means [latex]f\\left(-1\\right)=4[\/latex] and [latex]f\\left(3\\right)=4[\/latex], or when the input is [latex]-1[\/latex] or [latex]\\text{3,}[\/latex] the output is [latex]\\text{4}\\text{.}[\/latex] See the graph below.
            \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Graph of a parabola with two plotted points and a line intersecting both[\/caption]\r\n\r\n \r\n\r\n<\/center><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
            [ohm2_question hide_question_numbers=1]13515[\/ohm2_question]<\/section>
            You can use an online graphing calculator to graph functions, find function values, and evaluate functions. Watch this short tutorial to learn how to within Desmos. Other online graphing tools will be slightly different.\r\n