What is an Inverse?#
An inverse function is the mathematical “undo” button. If a function $f$ takes an input $x$ and produces an output $y$, then the inverse $f^{-1}$ takes $y$ as input and gives back the original $x$.
$$f: x \to y \qquad \text{then} \qquad f^{-1}: y \to x$$The “Machine” Analogy#
Think of $f$ as a machine:
- You feed in $x = 3$, and out comes $y = 10$
- The inverse machine $f^{-1}$ takes $10$ and gives back $3$
The inverse reverses the process — it “undoes” whatever the original function did.
The Formal Definition#
$f^{-1}$ is the inverse of $f$ if and only if:
$$f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x$$Applying the function and then its inverse (in either order) gets you back to where you started.
1. The “Swap” Rule — Finding an Inverse#
The Method#
To find the inverse of any function:
- Write the equation: $y = \dots$
- Swap $x$ and $y$: replace every $x$ with $y$ and every $y$ with $x$
- Solve for $y$ (make $y$ the subject)
- Write in inverse notation: $f^{-1}(x) = \dots$
Why Does Swapping Work?#
If the original function maps $x$ values to $y$ values, the inverse maps $y$ values back to $x$ values. Swapping $x$ and $y$ literally reverses the input-output relationship.
Worked Example 1 — Linear Inverse#
Find the inverse of $f(x) = 2x + 6$.
Step 1: $y = 2x + 6$
Step 2 — Swap: $x = 2y + 6$
Step 3 — Solve for $y$:
$$x - 6 = 2y$$$$y = \frac{x - 6}{2}$$Step 4: $f^{-1}(x) = \frac{x - 6}{2}$
Verify: $f(3) = 2(3) + 6 = 12$. Does $f^{-1}(12) = 3$? $\frac{12 - 6}{2} = 3$ ✓
Worked Example 2 — Exponential Inverse#
Find the inverse of $f(x) = 3^x$.
Step 1: $y = 3^x$
Step 2 — Swap: $x = 3^y$
Step 3 — Solve for $y$: Apply $\log_3$ to both sides:
$$y = \log_3 x$$Step 4: $f^{-1}(x) = \log_3 x$
This is why logarithms exist — they are the inverses of exponential functions.
2. The Graphical Relationship: Reflection Across $y = x$#
The graph of $f^{-1}$ is always a reflection of the graph of $f$ in the line $y = x$.
Why?#
If the point $(a; b)$ is on $f$, then $f(a) = b$. By definition, $f^{-1}(b) = a$, so the point $(b; a)$ is on $f^{-1}$.
Swapping coordinates $(a; b) \to (b; a)$ is exactly what reflection in $y = x$ does.
Worked Example 3 — Points on the Inverse#
The following points lie on $f$: $(1; 4)$, $(2; 7)$, $(3; 10)$. Write down points on $f^{-1}$.
Swap each coordinate:
| Point on $f$ | Point on $f^{-1}$ |
|---|---|
| $(1; 4)$ | $(4; 1)$ |
| $(2; 7)$ | $(7; 2)$ |
| $(3; 10)$ | $(10; 3)$ |
3. One-to-One vs Many-to-One#
Not every function can be inverted to give another function.
One-to-One Functions#
A function is one-to-one if every output comes from exactly one input. The horizontal line test confirms this: if any horizontal line crosses the graph more than once, the function is not one-to-one.
Examples of one-to-one functions:
- $y = mx + c$ (linear, $m \neq 0$)
- $y = a^x$ (exponential)
- $y = \log_a x$ (logarithmic)
These can be inverted directly — the inverse is automatically a function.
Many-to-One Functions#
A many-to-one function maps different inputs to the same output. The classic example is the parabola:
$$(-3)^2 = 9 \quad \text{and} \quad 3^2 = 9$$If you try to “undo” $9$, you get two answers: $-3$ and $3$. The inverse is not a function because one input ($9$) gives two outputs.
The Solution: Domain Restriction#
To make the inverse of a many-to-one function into a valid function, we restrict the domain of the original. For $y = x^2$:
- Restrict to $x \geq 0$: inverse is $y = \sqrt{x}$ (the right half)
- Restrict to $x \leq 0$: inverse is $y = -\sqrt{x}$ (the left half)
| Original | Domain restriction | Inverse |
|---|---|---|
| $y = x^2$ | $x \geq 0$ | $f^{-1}(x) = \sqrt{x}$ |
| $y = x^2$ | $x \leq 0$ | $f^{-1}(x) = -\sqrt{x}$ |
For the full treatment of each function type and its inverse, see the individual deep-dive pages: Linear, Quadratic, Exponential, Logarithmic.
4. Domain and Range Swap#
When you find an inverse, the domain and range swap:
| $f$ | $f^{-1}$ | |
|---|---|---|
| Domain | $\{x: x \in \mathbb{R}\}$ | = Range of $f$ |
| Range | $\{y: y > 0\}$ | = Domain of $f$ |
Worked Example 4 — Domain and Range#
$f(x) = 2^x$ has domain $x \in \mathbb{R}$ and range $y > 0$.
$f^{-1}(x) = \log_2 x$ has domain $x > 0$ and range $y \in \mathbb{R}$.
The domain and range have swapped — exactly as expected.
5. Verifying an Inverse#
To check that $g$ is truly the inverse of $f$, verify both compositions:
$$f(g(x)) = x \quad \text{and} \quad g(f(x)) = x$$Worked Example 5 — Verification#
$$f(g(x)) = 2\left(\frac{x-6}{2}\right) + 6 = (x - 6) + 6 = x \;\checkmark$$$$g(f(x)) = \frac{(2x + 6) - 6}{2} = \frac{2x}{2} = x \;\checkmark$$Verify that $f(x) = 2x + 6$ and $g(x) = \frac{x - 6}{2}$ are inverses.
Both compositions return $x$, confirming that $g = f^{-1}$.
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| $f^{-1}(x) = \frac{1}{f(x)}$ | The $-1$ is inverse notation, NOT a negative exponent | $f^{-1}$ means “undo $f$,” not “reciprocal of $f$” |
| Forgetting to swap $x$ and $y$ | Just solving for $x$ gives $x$ in terms of $y$ — not the inverse equation | Always swap first, then solve for $y$ |
| Not restricting the domain | The inverse of $y = x^2$ is $x = y^2$, which is not a function | Specify $x \geq 0$ or $x \leq 0$ before finding the inverse |
| Wrong reflection line | Reflecting in the $x$-axis or $y$-axis is not the same as $y = x$ | The inverse is a reflection in $y = x$ specifically |
| Domain/range confusion | The domain of $f^{-1}$ is the range of $f$ — not the same as the domain of $f$ | Always state both domain and range for $f$ and $f^{-1}$ |
💡 Pro Tips for Exams#
1. The “Point-Swap” Check#
If $(3; 10)$ is on the graph of $f$, then $(10; 3)$ must be on $f^{-1}$. This is the fastest way to check your inverse in an exam — substitute one known point.
2. The Line $y = x$ is Your Mirror#
When sketching both $f$ and $f^{-1}$ on the same axes, draw $y = x$ as a dashed line first. Every point on $f^{-1}$ should be the mirror image of the corresponding point on $f$ across this line.
3. Exponential ↔ Log Connection#
If the question involves $y = a^x$, the inverse is $y = \log_a x$ — and vice versa. You don’t need to go through the swap steps every time; just recognise the pair. See Logarithmic Function for the full treatment.