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Study Guides > College Algebra

Decompose a composite function into its component functions

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.

Example 10: Decomposing a Function

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


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

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

We can check our answer by recomposing the functions.


Try It 7

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

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..