Decomposing 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 \(f(x)=\sqrt{5-x^{2}}\) as the composition of two functions.


Solution

We are looking for two functions, \(g\) and \(h\), so \(f(x)=g(h(x))\). To do this, we look for a function inside a function in the formula for \(f(x)\). As one possibility, we might notice that the expression \(5-x^{2}\) is the inside of the square root. We could then decompose the function as

\(h(x)=5-x^{2} \quad \text { and } g(x)=\sqrt{x}\)

We can check our answer by recomposing the functions.

\(g(h(x))=g\left(5-x^{2}\right)=\sqrt{5-x^{2}}\)


TRY IT #7

Write \(f(x)=\dfrac{4}{3-\sqrt{4+x^{2}}}\) as the composition of two functions.