Composition of Functions¶

Definition¶

Let $f\colon X\to Y$ and $g\colon W\to Z$ be functions, and suppose that the range of $f$ is contained in the domain $W$ of $g\text{.}$ We define the composition $g\circ f\colon X\to Z$ by

Interpretation¶

Composition of functions describes how two functions can be combined so that the output of one becomes the input of the other. The composite $g\circ f$ is read from right to left: first apply $f$, then apply $g$ to the result.

The figure follows a single point: $f$ sends $x\in X$ to $f(x)\in W$$W$, and $g$ sends that point on to $g(f(x))\in Z$$Z$. The long arrow underneath skips the middle step and is the composite $g\circ f\colon X\to Z$$Z$. Both arrows out of $W$ end at the same point of $Z$, which is allowed: neither $g$ nor $g\circ f$ has to be injective.

The condition that the range of $f$ lies in the domain of $g$ makes $g(f(x))$ meaningful for every $x$. With $f(x)=x-1$ on $\mathbb{R}$ and $g(y)=\sqrt{y}$$\sqrt{y}$, the composite $\sqrt{x-1}$ only makes sense for $x\ge1$$1$, so $f$ must first be taken on $[1,\infty)$. Order matters as well: with $f(x)=x^2$ and $g(y)=\sin y$$\sin y$, $(g\circ f)(x)=\sin(x^2)$ but $(f\circ g)(x)=(\sin x)^2$$(\sin x{)}^2$, and at $x=\pi$ the first is $\sin(\pi^2)\approx-0.43$ while the second is $0$. A bijection composed with its inverse gives the identity function.

A common slip is to read $g\circ f$ as the product $g\cdot f$$f$. In the second Example $(g\circ f)(x)=2(x+1)$$1)$, whereas $g(x)f(x)=2x(x+1)$$1)$. Later, the chain rule gives the derivative of a composite.

Key Results¶

• The composition $g\circ f$ is defined when every output of $f$ lies in the domain of $g\text{,}$ that is, when the range of $f$ is contained in the domain of $g\text{.}$
• The actual condition is $f(X)\subseteq W$$W$, not equality of the declared middle sets $Y=W$$W$. Self-composition $f\circ f$ on all of $X$ requires $f(X)\subseteq X$$X$.
• Order matters: in general, $g\circ f$ and $f\circ g$ are different, and one may be defined even when the other is not.
• Composition preserves many structural properties, such as continuity and differentiability, under suitable conditions.

Examples¶

• If $f(x)=x^2$ and $g(y)=\sin y\text{,}$ then $(g\circ f)(x)=\sin(x^2)\text{.}$
• If $f(x)=x+1$ and $g(y)=2y\text{,}$ then $(g\circ f)(x)=2(x+1)\text{.}$

Prerequisites¶

• Function
• Domain
• Range and Image

Related Concepts¶

• Bijection and Inverse Function
• Identity Function
• Product of Linear Maps
• Function of a Random Variable
• Transformation of a Continuous Random Variable

Sources¶

• Math I: Calculus · Lecture 4

Aliases: function composition, composite function Symbols: element of, infinity, pi, real numbers, square root, subset

Last updated: 27 October 2026 11:52