# Compositions of Functions

Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function. For example, the function $2x^2$ can be thought of as the composition of two functions: the first function squares the input, and then the second function doubles the input.

Using formal notation, we can define the first function which squares the input as $f(x)=x^2$, and the second function which doubles the input as $g(x)=2x$. Then the composition can be computed by using the output of $f$ as the input to $g$. Starting at the end, we can compute the composition by evaluating in $g$ terms of $f$, and then evaluating $f$ in terms of $x$.

$(g \circ f)(x) = g(f(x)) = 2f(x) = 2x^2$

Or, we can start at the beginning, computing $f$ in terms of $x$ and then evaluating $g$ in terms of the result. Either way, we end up with the same formula for the composition.

$(g \circ f)(x) = g(f(x)) = g(x^2) = 2x^2$

The order of composition is very important and is not interchangeable. The function computed above is $g \circ f$, which applies $f$ first and then $g$. On the other hand, the function $f \circ g$ applies $g$ first and then $f$, and consequently evaluates to something different:$(f \circ g)(x) = 4x^2$.

For compositions of more than two functions, we can compute one step at a time.

\begin{align*} \text{Given functions} \hspace{.5cm} &\Bigg| \hspace{.5cm} f(x) = \sin x \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} g(x)=x^2 \\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} h(x)=5x+1\\ \text{ } \hspace{.5cm} &\Bigg| \hspace{.5cm} p(x)= \sqrt{x} \\ \text{Input } f \text{ into } g \hspace{.5cm} &\Bigg| \hspace{.5cm} (g \circ f)(x) = \sin^2 x \\ \text{Input } g \circ f \text{ into } h \hspace{.5cm} &\Bigg| \hspace{.5cm} (h \circ g \circ f)(x) = 5\sin^2 x + 1 \\ \text{Input } h \circ g \circ f \text{ into } p \hspace{.5cm} &\Bigg| \hspace{.5cm} (p \circ h \circ g \circ f)(x) = \sqrt{ 5\sin^2 x + 1 } \end{align*}

Practice Problems

Find the expression for the indicated composition. (You can view the solution by clicking on the problem.)

\begin{align*} 1) \hspace{.5cm} &(g \circ f)(x) = \text{___} \\ &f(x)=x+5 \\ &g(x)=2x^2 \end{align*}
Solution:
$2(x+5)^2$

\begin{align*} 2) \hspace{.5cm} &(g \circ f)(x) = \text{___} \\ &f(x)=5^x \\ &g(x)= |4-x| \end{align*}
Solution:
$|4-x^5|$

\begin{align*} 3) \hspace{.5cm} &(h \circ g \circ f)(x) = \text{___} \\ &f(x)=-2^x \\ &g(x)=|x+4| \\ &h(x) = \sqrt{x} \end{align*}
Solution:
$\sqrt{ |-2^x+4| }$

\begin{align*} 4) \hspace{.5cm} &(h \circ g \circ f)(x) = \text{___} \\ &f(x)=2x \\ &g(x)=\frac{x}{x-1} \\ &h(x)=\sin x \end{align*}
Solution:
$\sin \left( \frac{2x}{2x-1} \right)$

\begin{align*} 5) \hspace{.5cm} &(p \circ h \circ g \circ f)(x) = \text{___} \\ &f(x)=\sin x \\ &g(x)=x^2 \\ &h(x) = 1+\sqrt[3]{x} \\ &p(x) = \sqrt{x} \end{align*}
Solution:
$\sqrt{ 1+\sqrt[3]{ \sin^2 x } }$

\begin{align*} 6) \hspace{.5cm} &(p \circ h \circ g \circ f)(x) = \text{___} \\ &f(x)=\sqrt{x} \\ &g(x)=\tan x \\ &h(x) = \log_3 x \\ &p(x) = |x|^3 \end{align*}
Solution:
$\left| \log ( \tan ( \sqrt{x} )) \right|^3$