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

Solution

We are looking for two functions, $g\\$ and $h\\$ , so $f\left(x\right)=g\left(h\left(x\right)\right)\\$ . To do this, we look for a function inside a function in the formula for $f\left(x\right)\\$ . 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

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

We can check our answer by recomposing the functions.

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

Try It 7

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

Solution

Composition of Functions