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Exercises Functions

Example 10 from Algebra and Trigonometry, 3.4 Composition of Functions

Decomposing a Function

Write $h (x) = 5 - x^{2} and g (x) = \sqrt{x}$ as the composition of two functions.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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

$h (x) = 5 - x^{2} and g (x) = \sqrt{x}$
We can check our answer by recomposing the functions.

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

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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

How did it go?

PreviousFinding the Domain of a Composite Function Involving Radicals Next Adding a Constant to a Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.