Q.If $f : A \to B$, $g : B \to C$ are bijective functions then prove that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Function Operations: Combining Machines

Think of a function as a machine. You feed it an input (say, a number $x$), it does something, and out comes an output $f(x)$. Now imagine you have two such machines, $f$ and $g$. Function operations are simply ways to hook these machines together — to add, subtract, multiply, or divide their outputs, or to feed one machine's output into the other.

The core idea is simple: if you can do arithmetic with numbers, you can do arithmetic with functions. The only catch is that both functions must be "ready to work" on the same input at the same time.

The Four Arithmetic Operations

Let $f$ and $g$ be two functions. For any input $x$ that belongs to both their domains (the set of numbers each can accept), we define:

Example. Let $f(x) = \sqrt{x}$ (domain: $x \geq 0$) and $g(x) = x - 1$ (domain: all real numbers). Then:

• $(f+g)(x) = \sqrt{x} + x - 1$, domain: $x \geq 0$.
• $\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x-1}$, domain: $x \geq 0$ and $x \neq 1$.

Composition: Feeding One Machine into Another

This is the most powerful operation. Instead of adding outputs side by side, you take the output of one function and feed it as the input to the other.

Read "$f$ composed with $g$". You do $g$ first, then $f$ on the result.

Intuition. Suppose $g$ is a machine that converts Celsius to Fahrenheit, and $f$ is a machine that converts Fahrenheit to Kelvin. Then $f \circ g$ converts Celsius directly to Kelvin — one combined machine.

Domain trap. For $f(g(x))$ to make sense, two conditions must hold:

1. $x$ must be in the domain of $g$ (so $g(x)$ exists).
2. $g(x)$ must be in the domain of $f$ (so $f$ can accept it).

So the domain of $f \circ g$ is: all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.