MA001: College Algebra

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.