Composition of Functions and Invertible Function

Composition of functions (gof)

Let $f : A \to B$ and $g : B \to C$ be two functions. Their composition $g \circ f$ (read "$g$ of $f$") is the function $g \circ f : A \to C$ defined by

You apply $f$ first and feed its output into $g$; this requires the codomain of $f$ to be the domain of $g$. …

Invertible function (inverse f⁻¹)

A function $f : X \to Y$ is invertible if there is a function $g : Y \to X$ that reverses it:

where $I_X$ and $I_Y$ are the identity functions on $X$ and $Y$. Such a $g$ is unique; it is called the inverse of $f$ and denoted $f^{-1}$.

The two conditions say $g(f(x)) = x$ for all $x \in X$ and $f(g(y)) = y$ for all $y \in Y$ — each function undoes the other. A key fact: $f$ is invertible if and only if it is both one-one and onto. …

Fig 1.5 is a visual definition of function composition. It shows three sets, drawn as circles labelled A, B, and C, arranged left to right. Inside each circle sits one representative element: $x$ in A, $f(x)$ in B, and $g(f(x))$ in C. An arrow labelled $f$ arcs over the top from $x$ to $f(x)$. A second arrow labelled $g$ arcs from $f(x)$ to $g(f(x))$. A third, longer arrow labelled $g \circ f$ sweeps along the bottom directly from $x$ to $g(f(x))$, skipping the intermediate stop. The diagram makes the chain explicit: you first apply $f$, then $g$, and the composite arrow $g \circ f$ captures the entire journey in one step.

The physical idea is that composition is a two-stage process that you can treat as a single function. Given $f: A \to B$ and $g: B \to C$, the composite $g \circ f$ sends each $x$ in A directly to $g(f(x))$ in C. The figure emphasises that the output of $f$ becomes the input of $g$ — the middle set B is the bridge. Without the diagram, students often confuse the order: $g \circ f$ means "first $f$, then $g$", and the arrow from $x$ to $g(f(x))$ along the bottom makes that sequence visually unambiguous.

Here $f: A \to B$ and $g: B \to C$. The symbol $\circ$ is read as "composed with". The domain of $g \circ f$ is A, and its codomain is C. The formula works only when the range of $f$ lies inside the domain of $g$ — that is, every $f(x)$ must be a valid input for $g$. In the figure, that condition is satisfied because $f(x)$ sits inside B, which is exactly the domain of $g$.