Inverse functions

When an inverse exists

f^-1 exists only if f is one-one. The domain of f^-1 is the range of f, and the range of f^-1 is the domain of f. A many-one function such as x² needs its domain restricting (for example to x ≥ 0) first.

Finding f^-1(x)

1. Write y = f(x).

2. Rearrange to make x the subject.

3. Swap the letters: f^-1(x) = ...

4. State the domain of f^-1 (the range of f).

Graphs and facts

y = f^-1(x) is the reflection of y = f(x) in the line y = x.

ff^-1(x) = x and f^-1f(x) = x. For an increasing function, solutions of f(x) = f^-1(x) lie on y = x, so you can solve f(x) = x instead.

Worked example

f(x) = (2x + 3)/(x - 1), x > 1. Find f^-1(x).

1. y(x - 1) = 2x + 3
2. xy - y = 2x + 3, so x(y - 2) = y + 3
3. x = (y + 3)/(y - 2)
4. For x > 1, f(x) = 2 + 5/(x - 1) > 2, so the domain of the inverse is x > 2.

Answer: f^-1(x) = (x + 3)/(x - 2), x > 2

Check yourself

1. f(x) = 4x - 7. Find f^-1(x).
   • (x - 7)/4
   • (x + 7)/4
   • 4x + 7
   • 1/(4x - 7)
   Show answer

   (x + 7)/4. y = 4x - 7 gives x = (y + 7)/4.

2. f(x) = x³ + 2. Find f^-1(29).
   Show answer

   3. f^-1(29) is the x with x³ + 2 = 29, so x³ = 27 and x = 3.

3. f(x) = x² + 1, x ≥ 0. State the domain of f^-1.
   • x ≥ 0
   • x ≥ 1
   • x ∈ ℝ
   • x ≥ -1
   Show answer

   x ≥ 1. The range of f is f(x) ≥ 1, and this becomes the domain of f^-1(x) = sqrt(x - 1).

4. The graph of y = f^-1(x) is a reflection of y = f(x) in which line?
   • y = -x
   • The x-axis
   • y = x
   • The y-axis
   Show answer

   y = x. Swapping x and y reflects a graph in y = x.

5. Why does f(x) = x², x ∈ ℝ, have no inverse?
   • It is many-one: for example 4 comes from 2 and from -2
   • Its range is all real numbers
   • It is not continuous
   • Its graph does not pass through the origin
   Show answer

   It is many-one: for example 4 comes from 2 and from -2. An inverse would have to send 4 back to two values, which is not a function.

6. f(x) = 3x - 4. Solve f(x) = f^-1(x).
   Show answer

   2. f is increasing, so solve f(x) = x: 3x - 4 = x, giving x = 2. (Check: f^-1(x) = (x + 4)/3, and (2 + 4)/3 = 2.)

7. f(x) = (2x + 3)/(x - 1), x > 1. Without finding f^-1, write down the value of ff^-1(7).
   Show answer

   7. ff^-1(x) = x for every x in the domain of f^-1, so ff^-1(7) = 7.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.