Inverse Relations & Functions

Q: How are the graphs of $f$ and $f^{-1}$ related?

They're reflections of each other across the line $y = x$. Every point $(a, b)$ on $f$ corresponds to the point $(b, a)$ on $f^{-1}$. So if $f$ passes through $(2, 3)$, then $f^{-1}$ must pass through $(3, 2)$.

A function is a machine: put in an input, get out an output. The inverse function is the same machine run in reverse — it takes the output and hands back the original input. If $f(4) = 7$ , then the inverse, written $f^{-1}$ , satisfies $f^{-1}(7) = 4$ . Every input-output pair just trades places.

Inverses matter because most real questions come at a function backwards. A conversion formula turns Celsius into Fahrenheit — but you're handed Fahrenheit and want Celsius. A growth model turns time into population — but you know the population and want the time. Finding the inverse turns "solve for the input every single time" into a formula you use directly.

What an inverse function is

Formally, $f^{-1}$ is the function that reverses $f$ : whenever $f(a) = b$ , it's also true that $f^{-1}(b) = a$ . Every point $(a, b)$ on the graph of $f$ becomes the point $(b, a)$ on the graph of $f^{-1}$ . Inputs and outputs swap — that's the entire definition.

That coordinate swap has a visual consequence: the graph of $f^{-1}$ is the graph of $f$ reflected across the line $y = x$ . Fold the plane along that diagonal and the two curves land on each other. Below, a line (blue) and its inverse (red) are mirror images across the dashed line $y = x$ .

One caution on notation: the $-1$ in $f^{-1}(x)$ is not an exponent. $f^{-1}(x)$ means "the inverse function of $f$ ," not $\dfrac{1}{f(x)}$ . The inverse of $f(x) = x + 4$ is $f^{-1}(x) = x - 4$ , which is nothing like $\dfrac{1}{x + 4}$ .

How to find an inverse: swap and solve

The procedure is two moves. First, write the function as $y = \ldots$ and swap $x$ and $y$ — that's the input-output trade built into the definition. Second, solve the new equation for $y$ . What you get is $f^{-1}(x)$ .

For $f(x) = 2x + 6$ : write $y = 2x + 6$ , swap to get $x = 2y + 6$ , then solve — subtract $6$ to get $x - 6 = 2y$ , divide by $2$ to get $y = \dfrac{x}{2} - 3$ . So $f^{-1}(x) = \dfrac{x}{2} - 3$ .

Notice what the inverse does operation by operation: $f$ multiplies by $2$ and then adds $6$ ; $f^{-1}$ subtracts $6$ and then divides by $2$ . The inverse applies the opposite of each operation, in reverse order — the same way you take off shoes before socks.

The mapping below shows $f(x) = 2x + 6$ sending each input to its output. The inverse simply reverses every arrow: it reads the diagram right-to-left, sending $8 \to 1$ , $10 \to 2$ , and $12 \to 3$ .

How to check your inverse

Two functions are inverses exactly when composing them gets you back where you started: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ . Run any number through one machine and then the other, and it must come out unchanged. Check both directions — on a test, verifying composition is often the whole problem.

A faster spot-check: pick an easy input. If $f(1) = 8$ , then $f^{-1}(8)$ had better be $1$ . If it isn't, something went wrong in the swap-and-solve algebra.

One more wrinkle: not every function has an inverse that's a function. If $f$ sends two different inputs to the same output — like $f(x) = x^2$ sending both $2$ and $-2$ to $4$ — the reverse machine can't decide which input to return. Only one-to-one functions (graphs that pass the horizontal line test) have true inverse functions; that's why $x^2$ only gets an inverse after restricting it to $x \geq 0$ .

Worked examples

Example 1: a linear function

Find the inverse of $f(x) = 3x - 5$ .

Write with

y

y = 3x - 5

Swap

x

and

y

x = 3y - 5

Add

5

to both sides

x + 5 = 3y

Divide by

3

y = \dfrac{x + 5}{3}

Answer: $f^{-1}(x) = \dfrac{x + 5}{3}$ — check: $f(4) = 7$ and $f^{-1}(7) = \dfrac{12}{3} = 4$ ✓

Example 2: the variable is divided

Find the inverse of $f(x) = \dfrac{x}{2} + 1$ .

Write with

y

y = \dfrac{x}{2} + 1

Swap

x

and

y

x = \dfrac{y}{2} + 1

Subtract

1

from both sides

x - 1 = \dfrac{y}{2}

Multiply both sides by

2

y = 2x - 2

Answer: $f^{-1}(x) = 2x - 2$ — check: $f(6) = 4$ and $f^{-1}(4) = 6$ ✓

Example 3: verify two functions are inverses

Show that $f(x) = 2x + 6$ and $g(x) = \dfrac{x}{2} - 3$ are inverses.

Compose one direction: substitute

g

into

f

f(g(x)) = 2\left(\dfrac{x}{2} - 3\right) + 6

Distribute and simplify

= x - 6 + 6 = x

Compose the other direction: substitute

f

into

g

g(f(x)) = \dfrac{2x + 6}{2} - 3

Simplify

= x + 3 - 3 = x

Answer: Both compositions return $x$ , so $f$ and $g$ are inverses.

Example 4: a cubic function

Find the inverse of $f(x) = x^3 + 1$ .

Write with

y

and swap

x = y^3 + 1

Subtract

1

from both sides

x - 1 = y^3

Take the cube root of both sides

y = \sqrt[3]{x - 1}

Check with a point:

f(2) = 9

, and

f^{-1}(9) = \sqrt[3]{8} = 2

✓

Answer: $f^{-1}(x) = \sqrt[3]{x - 1}$

Try one yourself

Common questions

Does

f^{-1}(x)

mean

\dfrac{1}{f(x)}

No — this is the most common notation mistake with inverses. The $-1$ names the inverse function, not a reciprocal. The inverse of $f(x) = x + 4$ is $f^{-1}(x) = x - 4$ ; the reciprocal $\dfrac{1}{x + 4}$ is a completely different function.

Does every function have an inverse?

Every function can be reversed as a relation, but the result is only a function when the original is one-to-one — each output comes from exactly one input. Graphically, that's the horizontal line test. $f(x) = x^2$ fails it (both $2$ and $-2$ give $4$ ), so it only has an inverse function after you restrict its domain.

How are the graphs of

f

and

f^{-1}

related?

They're reflections of each other across the line $y = x$ . Every point $(a, b)$ on $f$ corresponds to the point $(b, a)$ on $f^{-1}$ . So if $f$ passes through $(2, 3)$ , then $f^{-1}$ must pass through $(3, 2)$ .

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Inverse Relations & Functions

What an inverse function is

How to find an inverse: swap and solve

How to check your inverse

Worked examples

Example 1: a linear function

Example 2: the variable is divided

Example 3: verify two functions are inverses

Example 4: a cubic function

Try one yourself

Common questions

Keep learning

Introduction to Functions

Logarithms & Logarithmic Functions

Transformations of Functions