Mathematics

What Is an Inverse Function?

An inverse undoes the original. Find the inverse of f(x) = 3x − 7 by swapping and solving, check it by substitution, and see why f⁻¹ is not 1 over f.

Inverse function

The function that reverses another, sending each output of f back to the input it came from, and written f⁻¹.

Also called: f inverse
Where students meet it: Grade 10 mathematics, on GCSE Higher tier and on the Extended papers of Cambridge IGCSE 0580, where functions are Extended content rather than Core.

The short answer
An inverse function undoes what the original did. If f(x) = 3x − 7 then f⁻¹(x) = (x + 7)/3, so f(5) = 8 and f⁻¹(8) = 5. Find it by writing y = f(x), swapping x and y, then making y the subject. The −1 is not a reciprocal.

An example

y = 3x − 7 → x = 3y − 7 → y = (x + 7)/3 → f⁻¹(x) = (x + 7)/3

Write the function as y = 3x − 7, swap every x with every y, then rearrange to get y on its own. Swapping is what performs the reversal; the rearranging is ordinary algebra afterwards. The check takes ten seconds and is worth doing every time: f(5) = 15 − 7 = 8, and f⁻¹(8) = 15/3 = 5. The input comes back, which is the only thing an inverse is required to do.

Swap, then solve

Three steps, always in the same order. Replace f(x) with y. Swap x and y throughout. Rearrange until y is alone, and rename it f⁻¹(x).

The swap is doing the mathematical work. A function takes an input to an output; its inverse takes that output back to the input; and exchanging the two letters is exactly how you write down that reversal. Everything after it is the same rearranging you would do to change the subject of any formula.

There is a second way of seeing the same answer, which is to reverse the operations one at a time. f(x) = 3x − 7 multiplies by 3 and then subtracts 7, so its inverse adds 7 and then divides by 3 — and (x + 7)/3 is precisely that. When the algebra gets tangled, this is a good check on whether the answer is plausible.

f⁻¹ does not mean one over f

The notation borrows a symbol that means something else everywhere in algebra, and the confusion it causes is entirely reasonable. In x⁻¹ the −1 means a reciprocal. In f⁻¹ it does not. It means the function that undoes f.

Numbers settle it. For f(x) = 3x − 7, the reciprocal 1/f(5) is 1/8 = 0.125, while the inverse f⁻¹(5) is 12/3 = 4. Two completely different values from the same starting point. The same clash appears with sin⁻¹, which is the angle whose sine is a given value, not 1/sin.

If a question genuinely wants a reciprocal it will write 1/f(x) or [f(x)]⁻¹, and the square brackets are the signal that the whole output is being inverted rather than the function reversed.

The functions that have no inverse

An inverse must return one value, so the original function must never send two inputs to the same output. f(x) = x² breaks this immediately: 3 and −3 both give 9, and an inverse asked about 9 would have no way of choosing. The rule can only be reversed once its domain is cut back, usually to x ≥ 0.

Graphically, the inverse is a reflection of the original in the line y = x, and this makes the failure visible. Reflect a parabola in y = x and you get a sideways parabola, which fails the vertical line test. Reflect any straight line that is not horizontal and you get another straight line, which is why every linear function apart from y = c has an inverse.

Common questions

Does f⁻¹(x) mean 1 divided by f(x)?

No. The −1 is notation for the inverse function, not an index. For f(x) = 3x − 7, f⁻¹(5) = 4 while 1/f(5) = 1/8. The two happen to share a symbol and share nothing else. A question wanting the reciprocal writes it as a fraction to avoid exactly this ambiguity.

How do you check an inverse is correct?

Feed a number through both. Pick any convenient input, apply f, then apply your f⁻¹ to the result. If the original number comes back, the inverse is right. Algebraically the same check is f⁻¹(f(x)) = x, which is worth doing whenever the rearranging involved more than one step.

Which functions do not have an inverse?

Any function where two different inputs share an output. Quadratics are the standard example: x² sends both 3 and −3 to 9, so reversing it is ambiguous. Restricting the domain fixes it. Functions that never repeat an output are called one-to-one, and those are precisely the invertible ones.

What does the graph of an inverse function look like?

The reflection of the original graph in the line y = x. Swapping x and y in the algebra is the same operation as swapping the axes in the picture, so every point (a, b) on f appears as (b, a) on f⁻¹. Sketching y = x first makes the reflection much easier to draw accurately.

Sources

Cambridge IGCSE Mathematics (0580) — Cambridge Assessment International Education

Last updated August 21, 2026