Foundations — Inverse functions — finding f⁻¹(x), horizontal line test

Before you can find the inverse or trust the horizontal line test, you need to read every mark on the page fluently. This note builds each symbol from nothing, in an order where each one leans only on the ones before it. Nothing here is assumed — if the parent note used it, we build it. (The notation $f^{-1}$ for the reverse rule is earned in §7; until then we just say "the reverse rule" in words.)

0. The starting picture: input → box → output

Everything begins with the idea of a function. A relaxed way to picture it is a machine: an arrow goes in on the left carrying a number, a labelled box does something to it, and an arrow comes out on the right carrying a new number.

But "machine" is just a picture. Here is the precise statement it stands for.

The whole chapter is careful bookkeeping about that rule and whether we can build a second rule that reverses it.

1. The symbol $x$ — the input

2. The symbol $y$ — the output

3. The notation $f$ and $f(x)$ — naming the box

So when we write $f(x)=3x-7$, that whole sentence says: the machine named $f$ takes an input, triples it, then subtracts $7$. Setting $y=f(x)$ just gives the output a short name, $y$.

4. Ordered pairs $(a,b)$ and the graph

Collect every such dot and you get the graph of $f$: a curve where each horizontal position $x$ has a dot sitting at height $f(x)$.

Why does this matter for inverses? Because the whole trick of the reverse rule is a picture move: if $(a,b)$ is on $f$, then $(b,a)$ is on the reverse rule. It just reads every dot backwards. Swapping the coordinates of a point is exactly reflecting it across the diagonal line $y=x$ — which is why the reverse rule's graph is $f$'s graph flipped over that diagonal. See Domain and Range for how this swap moves the allowed inputs and outputs.

5. Domain and range — the allowed inputs and outputs

The key fact for this chapter: the inverse swaps them.

$\text{domain of }f⟷\text{range of reverse rule},\text{range of }f⟷\text{domain of reverse rule}$

This is why $f(x)=\sqrt{x-2}$ (inputs $x\ge 2$, outputs $y\ge 0$) has reverse rule $x^2+2$ restricted to inputs $x\ge 0$. Full detail lives in Domain and Range.

6. Set notation: $A\to B$, $\in$, $\mathbb{R}$, and "onto"

A careful warning about reversing the arrow. It is tempting to write "reverse rule $:B\to A$," but this is only fully correct when $f$ is a bijection (defined in §8): one-to-one and onto. In general the reverse rule can only accept the values $f$ actually produces — its range, which sits inside $B$ but need not fill it. So the honest statement is:

$\text{reverse rule}:\left(\text{range of }f\right)\to A.$

For example $f(x)=\sqrt{x-2}$ has codomain $\mathbb{R}$ but range only $[0,\infty )$; its reverse rule accepts inputs $x\ge 0$, not every real. When $f$ is a bijection, range $=B$ and the tidy form $B\to A$ becomes exactly true.

7. The exponent notation $f^{-1}$ — the reverse machine

From here on we write $f^{-1}$ instead of "the reverse rule."

Chaining machines: the composition symbol $\circ$. To say "run $f$, then feed the result into $g$" we write $g\circ f$, read "==$g$ after $f$==", meaning $(g\circ f)(x)=g(f(x))$ — apply the inner machine first, then the outer. Notice the order reads right-to-left. The defining equations for the inverse are exactly two such chains, now that every symbol is earned:

Chaining two machines like this is called Function Composition — the operator $\circ$ is precisely the tool we use to verify an inverse.

8. One-to-one (injective), and bijection

Why this is the heart of the matter: if two inputs $2$ and $-2$ both produce $4$, then when the reverse machine is handed $4$ it cannot decide whether to return $2$ or $-2$. A machine that returns two answers for one input is not a function at all. So invertibility requires one-to-one.

9. The horizontal line test — reading one-to-one off the graph

First a symbol we will lean on: $⟺$ (read "if and only if") joins two statements that are true together or false together — each one guarantees the other. With that in hand:

Why it works, in one sentence: a horizontal line collects all points sharing one output value, so two hits mean two inputs sharing that output — exactly the failure of one-to-one.

The parabola on the left fails (the line hits twice); the cube-ish curve on the right passes (never more than one hit).

How these feed the topic

Read the map bottom-up: each box names a foundation, and the arrows show which idea depends on which.

So the swap method (the actual skill) needs the reverse-machine idea, which needs a bijection — one-to-one plus onto — which rests on the one-to-one condition, read off the graph via the horizontal line test, which itself needs the graph, pairs, and the meanings of $x$, $y$, and $f$.

Equipment checklist

Cover the right side and answer before revealing.

Next: with the vocabulary secured, move to the parent topic note and the swap-method worked examples.