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أدلة الدراسة > College Algebra

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(x)=5x2f\left(x\right)=\sqrt{5-{x}^{2}}\\ as the composition of two functions.

Solution

We are looking for two functions, gg\\ and hh\\, so f(x)=g(h(x))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(x)f\left(x\right)\\. As one possibility, we might notice that the expression 5x25-{x}^{2}\\ is the inside of the square root. We could then decompose the function as

h(x)=5x2 and g(x)=xh\left(x\right)=5-{x}^{2}\text{ and }g\left(x\right)=\sqrt{x}\\

We can check our answer by recomposing the functions.

g(h(x))=g(5x2)=5x2g\left(h\left(x\right)\right)=g\left(5-{x}^{2}\right)=\sqrt{5-{x}^{2}}\\

Try It 7

Write f(x)=434+x2f\left(x\right)=\frac{4}{3-\sqrt{4+{x}^{2}}}\\ as the composition of two functions. Solution

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