# What are some examples of function composition?

Dec 22, 2016

To compose a function is to input one function into the other to form a different function. Here's a few examples.

Example 1: If $f \left(x\right) = 2 x + 5$ and $g \left(x\right) = 4 x - 1$, determine $f \left(g \left(x\right)\right)$

This would mean inputting $g \left(x\right)$ for $x$ inside $f \left(x\right)$.

$f \left(g \left(x\right)\right) = 2 \left(4 x - 1\right) + 5 = 8 x - 2 + 5 = 8 x + 3$

Example 2: If $f \left(x\right) = 3 {x}^{2} + 12 + 12 x$ and $g \left(x\right) = \sqrt{3 x}$, determine $g \left(f \left(x\right)\right)$ and state the domain

Put $f \left(x\right)$ into $g \left(x\right)$.

$g \left(f \left(x\right)\right) = \sqrt{3 \left(3 {x}^{2} + 12 x + 12\right)}$

$g \left(f \left(x\right)\right) = \sqrt{9 {x}^{2} + 36 x + 36}$

$g \left(f \left(x\right)\right) = \sqrt{{\left(3 x + 6\right)}^{2}}$

$g \left(f \left(x\right)\right) = | 3 x + 6 |$

The domain of $f \left(x\right)$ is $x \in \mathbb{R}$. The domain of $g \left(x\right)$ is $x > 0$. Hence, the domain of $g \left(f \left(x\right)\right)$ is $x > 0$.

Example 3: if $h \left(x\right) = {\log}_{2} \left(3 {x}^{2} + 5\right)$ and $m \left(x\right) = \sqrt{x + 1}$, find the value of $h \left(m \left(0\right)\right)$?

Find the composition, and then evaluate at the given point.

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 {\left(\sqrt{x + 1}\right)}^{2} + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 \left(x + 1\right) + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 x + 3 + 5\right)$

$h \left(m \left(x\right)\right) = {\log}_{2} \left(3 x + 8\right)$

$h \left(m \left(2\right)\right) = {\log}_{2} \left(3 \left(0\right) + 8\right)$

$h \left(m \left(2\right)\right) = {\log}_{2} 8$

$h \left(m \left(2\right)\right) = 3$

Practice exercises

For the following exercises: $f \left(x\right) = 2 x + 7 , g \left(x\right) = {2}^{x - 7} \mathmr{and} h \left(x\right) = 2 {x}^{3} - 4$

a) Determine $f \left(g \left(x\right)\right)$

b) Determine $h \left(f \left(x\right)\right)$

c) Determine $g \left(h \left(2\right)\right)$

Hopefully this helps, and good luck!