Is f⁻¹(x) the same as 1/f(x)?

No. The −1 means the inverse function, the rule that undoes f. The reciprocal 1/f(x) is a different thing, and confusing the two is the most common error on this topic.

Does every function have an inverse?

Only a one-one function, where no two inputs share an output. A quadratic like x² is not one-one, so exam questions restrict its domain first, for example to x ≥ 0.

What is the domain of the inverse?

The range of the original function. The inverse takes the outputs of f as its inputs.

How to Find the Inverse of a Function (with Fractions)

Short answer

How do you find the inverse of a function?

Write y = f(x), rearrange to make x the subject, then swap x and y. For f(x) = 3x − 5 that gives f⁻¹(x) = (x + 5)/3. Check your answer: f(f⁻¹(x)) must give x back.

An inverse function undoes the original. If $f$ turns 2 into 1, then $f^{- 1}$ turns 1 back into 2.

The method, in three steps

Write $y = f (x)$ .
Rearrange to make $x$ the subject.
Swap $x$ and $y$ , and write $f^{- 1} (x)$ .

Example 1: a linear function

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

y = 3 x - 5 ⟹ y + 5 = 3 x ⟹ x = \frac{y + 5}{3}

Swap the letters:

f^{- 1} (x) = \frac{x + 5}{3}

Check: $f (4) = 7$ and $f^{- 1} (7) = \frac{12}{3} = 4$ . The inverse sends 7 back to 4.

Example 2: a function with a fraction

Find the inverse of $h (x) = \frac{2 x + 1}{x - 3}$ for $x \neq = 3$ .

Here $x$ appears twice, so the trick is to collect every $x$ term on one side and factorise.

y (x - 3) = 2 x + 1 ⟹ x y - 3 y = 2 x + 1 ⟹ x y - 2 x = 3 y + 1

x (y - 2) = 3 y + 1 ⟹ x = \frac{3 y + 1}{y - 2}

h^{- 1} (x) = \frac{3 x + 1}{x - 2}, x \neq = 2

Check: $h (4) = \frac{9}{1} = 9$ and $h^{- 1} (9) = \frac{28}{7} = 4$ .

Try it: move the input and watch the inverse undo it

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Mistakes that cost marks

Writing $f^{- 1} (x) = \frac{1}{3 x - 5}$ . That is the reciprocal, not the inverse.
Forgetting to swap $x$ and $y$ at the end, so the answer is left in terms of $y$ .
Not factorising out $x$ when it appears twice. Get every $x$ term on one side first.

Want exam questions on this? See 9709 functions practice.

How to find the inverse of a function