{"id":912,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=912"},"modified":"2015-11-12T18:37:58","modified_gmt":"2015-11-12T18:37:58","slug":"decompose-a-composite-function-into-its-component-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/decompose-a-composite-function-into-its-component-functions\/","title":{"raw":"Decompose a composite function into its component functions","rendered":"Decompose a composite function into its component functions"},"content":{"raw":"

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<\/span>, so we may choose the decomposition that appears to be most expedient.<\/p>\n\n

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Example 10: Decomposing a Function<\/h3>\n

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

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Solution<\/h3>\n

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<\/p>\n\n

[latex]h\\left(x\\right)=5-{x}^{2}\\text{ and }g\\left(x\\right)=\\sqrt{x}\\\\[\/latex]<\/div>\n

We can check our answer by recomposing the functions.<\/p>\n\n

[latex]g\\left(h\\left(x\\right)\\right)=g\\left(5-{x}^{2}\\right)=\\sqrt{5-{x}^{2}}\\\\[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Try It 7<\/h3>\nWrite [latex]f\\left(x\\right)=\\frac{4}{3-\\sqrt{4+{x}^{2}}}\\\\[\/latex] as the composition of two functions.\n\nSolution<\/a>\n\n<\/div>","rendered":"

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<\/span>, so we may choose the decomposition that appears to be most expedient.<\/p>\n

\n
\n
\n

Example 10: Decomposing a Function<\/h3>\n

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

\n

Solution<\/h3>\n

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<\/p>\n

[latex]h\\left(x\\right)=5-{x}^{2}\\text{ and }g\\left(x\\right)=\\sqrt{x}\\\\[\/latex]<\/div>\n

We can check our answer by recomposing the functions.<\/p>\n

[latex]g\\left(h\\left(x\\right)\\right)=g\\left(5-{x}^{2}\\right)=\\sqrt{5-{x}^{2}}\\\\[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Try It 7<\/h3>\n

Write [latex]f\\left(x\\right)=\\frac{4}{3-\\sqrt{4+{x}^{2}}}\\\\[\/latex] as the composition of two functions.<\/p>\n

Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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