{"id":1576,"date":"2024-05-28T22:46:14","date_gmt":"2024-05-28T22:46:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1576"},"modified":"2024-12-05T18:13:36","modified_gmt":"2024-12-05T18:13:36","slug":"combinations-and-compositions-of-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/combinations-and-compositions-of-functions-learn-it-4\/","title":{"raw":"Combinations and Compositions of Functions: Learn It 4","rendered":"Combinations and Compositions of Functions: Learn It 4"},"content":{"raw":"

Decomposing a Composite Function<\/h2>\r\nIn some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.\r\n\r\n
Write [latex]f(x)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.<\/strong>\r\n\r\n
\r\n\r\n\r\n<\/strong>We are looking for two functions, [latex]g[\/latex] and [latex]h[\/latex], so [latex]f\\left(x\\right)=g\\left(h\\left(x\\right)\\right)[\/latex]. To do this, we look for a function inside a function in the formula for [latex]f\\left(x\\right)[\/latex].There are multiple ways to express [latex]f(x)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.[reveal-answer q=\"53139\"]Option 1[\/reveal-answer]\r\n[hidden-answer a=\"53139\"]\r\n
    \r\n \t
  • [latex]h(x) = 5 - x^2[\/latex]<\/li>\r\n \t
  • [latex]g(x) = \\sqrt{x}[\/latex]<\/li>\r\n<\/ul>\r\nThus:\r\n\r\n
    [latex]\\begin{align*} f(x) = g(h(x)) &= g(5 - x^2) \\\\ \\text{Since } g(x) &= \\sqrt{x}, \\text{ we have:} \\\\ g(5 - x^2) &= \\sqrt{5 - x^2} \\end{align*}[\/latex]<\/center>\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"80338\"]Option 2[\/reveal-answer]\r\n[hidden-answer a=\"80338\"]\r\n
      \r\n \t
    • [latex]h(x) = x^2[\/latex]<\/li>\r\n \t
    • [latex]g(x) = \\sqrt{5 - x}[\/latex]<\/li>\r\n<\/ul>\r\nThus:\r\n\r\n
      [latex]\\begin{align*} f(x) = g(h(x)) &= g(x^2) \\\\ \\text{Since } g(x) &= \\sqrt{5 - x}, \\text{ we have:} \\\\ g(x^2) &= \\sqrt{5 - x^2} \\end{align*}[\/latex]<\/center>\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"399617\"]Option 3[\/reveal-answer]\r\n[hidden-answer a=\"399617\"]\r\n
        \r\n \t
      • [latex]h(x) = x[\/latex]<\/li>\r\n \t
      • [latex]g(x) = \\sqrt{5 - x^2}[\/latex]<\/li>\r\n<\/ul>\r\nThus:\r\n\r\n
        [latex]\\begin{align*} f(x) = g(h(x)) &= g(x) \\\\ \\text{Since } g(x) &= \\sqrt{5 - x^2}, \\text{ we have:} \\\\ g(x) &= \\sqrt{5 - x^2} \\end{align*}[\/latex]<\/center>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
        Write [latex]f\\left(x\\right)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.[reveal-answer q=\"702975\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"702975\"]We are looking for two functions, [latex]g[\/latex] and [latex]h[\/latex], so [latex]f\\left(x\\right)=g\\left(h\\left(x\\right)\\right)[\/latex]. To do this, we look for a function inside a function in the formula for [latex]f\\left(x\\right)[\/latex]. As one possibility, we might notice that the expression [latex]5-{x}^{2}[\/latex] is the inside of the square root. We could then decompose the function as\r\n

        [latex]h\\left(x\\right)=5-{x}^{2}\\hspace{2mm}\\text{and}\\hspace{2mm}g\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\nWe can check our answer by recomposing the functions.\r\n

        [latex]g\\left(h\\left(x\\right)\\right)=g\\left(5-{x}^{2}\\right)=\\sqrt{5-{x}^{2}}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nFor every composition there are infinitely many possible function pairs that will work. In this case, another function pair where\u00a0[latex]g\\left(h\\left(x\\right)\\right)=\\sqrt{5-{x}^{2}}[\/latex]\u00a0 is\u00a0 [latex]h(x)=x^2[\/latex] and [latex]g(x)=\\sqrt{5-x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        [ohm2_question hide_question_numbers=1]19138[\/ohm2_question]<\/section>
        [ohm2_question hide_question_numbers=1]19139[\/ohm2_question]<\/section>","rendered":"

        Decomposing a Composite Function<\/h2>\n

        In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.<\/p>\n

        Write [latex]f(x)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.<\/strong><\/p>\n
        \n


        \n<\/strong>We are looking for two functions, [latex]g[\/latex] and [latex]h[\/latex], so [latex]f\\left(x\\right)=g\\left(h\\left(x\\right)\\right)[\/latex]. To do this, we look for a function inside a function in the formula for [latex]f\\left(x\\right)[\/latex].There are multiple ways to express [latex]f(x)=\\sqrt{5-{x}^{2}}[\/latex] as the composition of two functions.<\/p>\n