What is the inverse of f(x) = 5x?

f⁻¹(x) = x/5. Multiplying by 5 is undone by dividing by 5.

The graph of y = f⁻¹(x) is the reflection of y = f(x) in which line?

y = x. The x and y coordinates of every point swap.

Why does f(x) = x², for all real x, have no inverse?

It is many–one. Two inputs give the same output, so the output cannot be traced back to one input.

What is the inverse of f(x) = 2x − 6?

f⁻¹(x) = (x + 6)/2. Add 6, then divide by 2.

Composite and inverse functions · Cambridge IGCSE Additional Mathematics

In plain words

Two machines can be joined: the output of one becomes the input of the other. That is a composite function. And a machine can be run backwards, if it is the right kind: that is an inverse.

7 things to know

fg(x) means f(g(x)): do g first, then f. The order matters, and fg is usually different from gf.
f²(x) means ff(x): apply f twice.
The inverse function f⁻¹ undoes f. Only a one–one function has an inverse.
To find f⁻¹: write y = f(x), rearrange to make x the subject, then swap the letters.
The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
The graph of y = f⁻¹(x) is the reflection of the graph of y = f(x) in the line y = x.
The functions eˣ and ln x are inverses of each other.

Worked example

f(x) = 2x + 3 and g(x) = x². Find fg(x), gf(x) and fg(2).

fg(x) = f(x²) = 2x² + 3.
gf(x) = g(2x + 3) = (2x + 3)².
fg(2) = 2 × 4 + 3 = 11. Note that gf(2) = 7² = 49: the order matters.

Worked example

Find the inverse of f(x) = (x + 1)/(x − 2), where x ≠ 2.

Write y = (x + 1)/(x − 2). Multiply up: y(x − 2) = x + 1.
Expand and collect the x terms: xy − 2y = x + 1, so xy − x = 2y + 1, so x(y − 1) = 2y + 1.
x = (2y + 1)/(y − 1). Swap the letters: f⁻¹(x) = (2x + 1)/(x − 1), where x ≠ 1.

Tips and tricks

In fg(x), the function nearest the x is used first. Read it from right to left.
Write f⁻¹(x) = … in terms of x. Leaving the answer as "x = …" in terms of y loses the last mark.

6 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Composite and inverse functions: 6 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

f(x) = x + 1 and g(x) = x². What is fg(3)?
- 7
- 10 (the answer)
- 16
- 12
g(3) = 9, then f(9) = 10.
f(x) = x + 1 and g(x) = x². What is gf(3)?
- 10
- 12
- 16 (the answer)
- 9
f(3) = 4, then g(4) = 16.
What is the inverse of f(x) = 5x?
- f⁻¹(x) = 5/x
- f⁻¹(x) = x/5 (the answer)
- f⁻¹(x) = x − 5
- f⁻¹(x) = −5x
Multiplying by 5 is undone by dividing by 5.
The graph of y = f⁻¹(x) is the reflection of y = f(x) in which line?
- the x-axis
- the y-axis
- y = x (the answer)
- y = −x
The x and y coordinates of every point swap.
Why does f(x) = x², for all real x, have no inverse?
- It is not a function.
- It is many–one. (the answer)
- Its range is too small.
- It has no domain.
Two inputs give the same output, so the output cannot be traced back to one input.
What is the inverse of f(x) = 2x − 6?
- f⁻¹(x) = 2x + 6
- f⁻¹(x) = (x − 6)/2
- f⁻¹(x) = (x + 6)/2 (the answer)
- f⁻¹(x) = x/2 + 6
Add 6, then divide by 2.

6 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.