Composition of Functions

Notation	Meaning
$(f\circ g)(x)$	$f(g(x))$: apply $g$ first, then $f$
Domain of $f\circ g$	All $x$ in Dom($g$) such that $g(x)\in$ Dom($f$)

Building a Composition

Let $f(x)=x^2+1$ and $g(x)=1/x$. We can define a new function $h$ by applying $g$ first, then $f$:

$$h(x)=f(g(x))=f\left(\frac{1}{x}\right)={\left(\frac{1}{x}\right)}^2+1=\frac{1}{x^2}+1=\frac{x^2+1}{x^2}.$$

To obtain $h(x)$, we substituted $g(x)$ in place of $x$ in the expression for $f(x)$.

The General Definition

Suppose $f$ and $g$ are two given functions. Starting with $x$ in the domain of $g$, we apply $g$ to get $g(x)$, then apply $f$ to $g(x)$ to get $f(g(x))$.

The new function that takes $x$ and assigns $f(g(x))$ is called the composition of $f$ and $g$, denoted $f\circ g$. The symbol $f\circ g$ is read "$f$ circle $g$."

This composition is defined only when $g(x)$ is in the domain of $f$, otherwise, we cannot use $g(x)$ as the input for $f$.

Order Matters

In general, $f\circ g$ is not the same as $g\circ f$. For example, if $f(x)=\sin x$ and $g(x)=\sqrt{x+1}$:

$$f\circ g(x)=f(g(x))=f(\sqrt{x+1})=\sin (\sqrt{x+1})$$

but

$$g\circ f(x)=g(f(x))=g(\sin x)=\sqrt{1+\sin x}.$$

These are clearly different functions with different domains.

Decomposing Functions

It is equally important to recognize a given function as a composition of simpler pieces. For example, $h(x)=\sin (x^2)$ is $f\circ g$ where $f(x)=\sin x$ and $g(x)=x^2$. Identifying the inner and outer functions is the first step in applying the chain rule.

Frequently Asked Questions

Can I compose more than two functions?

Yes. $f\circ g\circ h$ means "apply $h$ first, then $g$, then $f$": $(f\circ g\circ h)(x)=f(g(h(x)))$. Composition is associative: $(f\circ g)\circ h=f\circ (g\circ h)$, so the grouping does not matter, only the order.

How does composition connect to the chain rule?

The chain rule says: if $h(x)=f(g(x))$, then h'(x) = f'(g(x)) \cdot g'(x). The derivative of a composition is the derivative of the outer function (evaluated at the inner function) times the derivative of the inner function. Identifying the inner and outer functions of a composition is the key skill for applying the chain rule correctly.

What is the domain of a composition?

The domain of $f\circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$. For example, if $f(x)=\sqrt{x}$ and $g(x)=1-x^2$, then $(f\circ g)(x)=\sqrt{1-x^2}$. For this to be defined, we need $1-x^2\ge 0$, i.e., $-1\le x\le 1$. Domain: $[-1,1]$.