{"id":3429,"date":"2025-08-20T20:40:23","date_gmt":"2025-08-20T20:40:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=3429"},"modified":"2026-01-08T17:07:28","modified_gmt":"2026-01-08T17:07:28","slug":"composition-of-functions-learn-it-4-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/composition-of-functions-learn-it-4-2\/","title":{"raw":"Composition of Functions: Learn It 4","rendered":"Composition of Functions: Learn It 4"},"content":{"raw":"

Evaluating Composite Functions from Formulas<\/h2>\r\n
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\r\n\r\nWhen evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.\r\n\r\nWhile we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition [latex](f \\circ g)(x)[\/latex].\r\n\r\n
How To: Given a formula for a composite function, evaluate the function.\r\n<\/strong>\r\n
    \r\n \t
  1. Evaluate the inside function using the input value or variable provided.<\/li>\r\n \t
  2. Use the resulting output as the input to the outside function.<\/li>\r\n<\/ol>\r\n<\/section>
    Given that [latex]f(x) = x^2-x[\/latex] and [latex]h(x) = 3x+2[\/latex], find [latex](f \\circ h)(1)[\/latex], [latex](h \\circ f)(1)[\/latex], and [latex](f \\circ h \\circ f)(1)[\/latex].[reveal-answer q=\"45744\"][latex](f \\circ h)(1)[\/latex][\/reveal-answer]\r\n[hidden-answer a=\"45744\"]
    [latex]\\begin{align*} \\text{First, find } h(1): \\\\ h(1) &= 3(1) + 2 \\\\ &= 3 + 2 \\\\ &= 5 \\\\[2mm] \\text{Next, find } f(h(1)) = f(5): \\\\ f(5) &= 5^2 - 5 \\\\ &= 25 - 5 \\\\ &= 20 \\\\[2mm] \\text{Therefore, } f(h(1)) &= 20 \\end{align*}[\/latex]<\/center>We can also find the composition function [latex]f \\circ h[\/latex] first, and then use this to to evaluate the expressions.
    [latex]\\begin{align*} (f \\circ h)(x) &= f(h(x)) \\\\ &= f(3x + 2) \\\\ &= (3x + 2)^2 - (3x + 2) \\\\ &= 9x^2 + 12x + 4 - 3x - 2 \\\\ &= 9x^2 + 9x + 2 \\end{align*}[\/latex]<\/center>Therefore,
    [latex](f \\circ h)(1) = 9(1)^2 + 9(1) + 2 = 9+9+2 = 20[\/latex]<\/center>.[\/hidden-answer][reveal-answer q=\"641220\"][latex](h \\circ f)(1)[\/latex][\/reveal-answer]\r\n[hidden-answer a=\"641220\"]\\begin{align*} \\text{First, find } f(1): \\\\ f(1) &= 1^2 - 1 \\\\ &= 1 - 1 \\\\ &= 0 \\\\[2mm] \\text{Next, find } h(f(1)) = h(0): \\\\ h(0) &= 3(0) + 2 \\\\ &= 0 + 2 \\\\ &= 2 \\\\[2mm] \\text{Therefore, } h(f(1)) &= 2 \\end{align*}[\/hidden-answer][reveal-answer q=\"240587\"][latex](f \\circ h \\circ f)(1)[\/latex][\/reveal-answer]\r\n[hidden-answer a=\"240587\"]\\begin{align*} \\text{First, find } f(1): \\\\ f(1) &= 1^2 - 1 \\\\ &= 1 - 1 \\\\ &= 0 \\\\[2mm] \\text{Next, find } h(f(1)) = h(0): \\\\ h(0) &= 3(0) + 2 \\\\ &= 0 + 2 \\\\ &= 2 \\\\[2mm] \\text{Finally, find } f(h(f(1))) = f(2): \\\\ f(2) &= 2^2 - 2 \\\\ &= 4 - 2 \\\\ &= 2 \\\\[2mm] \\text{Therefore, } f(h(f(1))) &= 2 \\end{align*}[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]317904[\/ohm_question]<\/section>
    When evaluating the composition of functions at a specific value, always check the domains of both functions first. The value must be in the domain of the inner function, and the result must be in the domain of the outer function. This ensures the composition is valid.<\/section>
    What value(s) of [latex]x[\/latex] that are not allowed for the composition [latex]f \\circ g[\/latex] if [latex]f(x) = \\dfrac{5}{x - 1}[\/latex] and [latex]g(x) = \\dfrac{4}{3x - 2}[\/latex]?[reveal-answer q=\"845324\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"845324\"]Check the domain of [latex]g(x)[\/latex]:\r\n
      \r\n \t
    • [latex]g(x) = \\dfrac{4}{3x - 2} \\text{ is defined for } 3x - 2 \\neq 0.[\/latex]<\/li>\r\n \t
    • [latex]3x - 2 \\neq 0 \\Rightarrow x \\neq \\dfrac{2}{3}.[\/latex]<\/li>\r\n<\/ul>\r\nCheck the domain of [latex]f(g(x))[\/latex]:\r\n
        \r\n \t
      • [latex]f(x) = \\dfrac{5}{x - 1} \\text{ is defined for } x - 1 \\neq 0 \\Rightarrow x \\neq 1.[\/latex]<\/li>\r\n \t
      • Our input is now [latex]g(x)[\/latex]. So, we need to ensure that [latex]g(x) \\neq 1[\/latex].<\/li>\r\n \t
      • [latex]g(x) = \\dfrac{4}{3x - 2} \\neq 1 \\Rightarrow \\dfrac{4}{3x - 2} \\neq 1 \\Rightarrow 4 \\neq 3x - 2 \\Rightarrow 3x \\neq 6 \\Rightarrow x \\neq 2[\/latex]<\/li>\r\n<\/ul>\r\nTherefore, the values of [latex]x[\/latex] that are not allowed are [latex]x \\neq \\dfrac{2}{3}[\/latex] and [latex]x \\neq 2[\/latex].\r\n\r\nNote: The value [latex]x=1[\/latex] does not need to be excluded since it is not directly applicable in this context; [latex]x=1[\/latex] would be relevant if we were evaluating [latex]f(x)[\/latex] directly.[\/hidden-answer]\r\n\r\n<\/section>
        [ohm_question hide_question_numbers=1]317905[\/ohm_question]<\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"

        Evaluating Composite Functions from Formulas<\/h2>\n
        \n
        \n
        \n

        When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.<\/p>\n

        While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition [latex](f \\circ g)(x)[\/latex].<\/p>\n

        How To: Given a formula for a composite function, evaluate the function.
        \n<\/strong><\/p>\n
          \n
        1. Evaluate the inside function using the input value or variable provided.<\/li>\n
        2. Use the resulting output as the input to the outside function.<\/li>\n<\/ol>\n<\/section>\n
          Given that [latex]f(x) = x^2-x[\/latex] and [latex]h(x) = 3x+2[\/latex], find [latex](f \\circ h)(1)[\/latex], [latex](h \\circ f)(1)[\/latex], and [latex](f \\circ h \\circ f)(1)[\/latex].<\/p>\n