Combinations and Compositions of Functions: Learn It 4
Decomposing a Composite Function
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.
Write [latex]f(x)=\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].There are multiple ways to express [latex]f(x)=\sqrt{5-{x}^{2}}[/latex] as the composition of two functions.
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
For every composition there are infinitely many possible function pairs that will work. In this case, another function pair where [latex]g\left(h\left(x\right)\right)=\sqrt{5-{x}^{2}}[/latex] is [latex]h(x)=x^2[/latex] and [latex]g(x)=\sqrt{5-x}[/latex]