Functions are machines

The idea

A function turns each allowed input into exactly one output. Think of $f(x)$ as a named machine: something goes in, a fixed rule is applied, and exactly one result comes out every time.

Composition chains machines together: $(g \circ f)(x)$ means run $f$ first, then feed its output into $g$.

An inverse function runs a reversible machine backward. To find $f^{-1}$, write $y=f(x)$ and solve for $x$. For $f(x)=3x-2$, $y=3x-2$ gives $x=(y+2)/3$, so $f^{-1}(y)=(y+2)/3$. An inverse exists on a domain only when each output identifies one input; otherwise the domain must be restricted.

Real-world: a ride-fare calculator

A taxi app charges a $3 booking fee plus $2 per kilometre; distance goes in and total fare comes out, so the rule is $f(x) = 2x + 3$. A 6 km trip gives $f(6) = 2(6) + 3 = 15$, i.e. $15 — one distance always produces one fare.

Worked example

1. Input $x = 3$
2. Multiply by 2: $6$
3. Add 1: $7$
4. So $f(3) = 7$

Cheat sheet

$f(x) = 2x + 1$

• $f(x)$ names a fixed rule: the same input always gives the same output.
• Composition $(g \circ f)(x)$ runs $f$ first, then applies $g$ to the result.
• An inverse reverses a one-to-one rule: solve $y=f(x)$ for $x$, then verify $f^{-1}(f(x))=x$.
• A function may send two different inputs to the same output, but never one input to two outputs.
• To evaluate $f$ at a value, substitute it for every $x$ in the rule and simplify.