72 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.
Solution
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
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.
