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Grade 9 · Higher · free GCSE maths worksheet

Inverse of a composite function

Combining f, g and their inverses.

9 original questions, 40 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

f(x) = 2x + 1   and   g(x) = x − 4

Find (fg)−1(x).

3 marks
Show answer
(x + 7)/2
2

f(x) = 3x − 2   and   g(x) = x + 5

Show that (fg)−1(x) = g−1f−1(x).

4 marks
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Both sides equal (x - 13)/3
3

f(x) = x + 3    g(x) = 4x

  1. Find fg(x)

  2. Find (fg)−1(x)

  3. Find (fg)−1(11)

4 marks
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a) 4x + 3 b) (x - 3)/4 c) 2
4

f(x) = 5x − 3    g(x) = x + 12

  1. Find gf(1)

  2. Find g−1f(2)

4 marks
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a) 3/2 b) 13
5

f(x) = 2x + 1x − 3, x ≠ 3

  1. Show that f−1(x) = 3x + 1x − 2

  2. g(x) = x + 4

    Solve fg(x) = 5

6 marks
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a) Rearranging y = (2x + 1)/(x - 3) gives x = (3y + 1)/(y - 2), so f-1(x) = (3x + 1)/(x - 2) b) x = 4/3
6

f(x) = x + 2    g(x) = x2 − 3x

  1. Find fg(1)

  2. Find f−1(x)

  3. Solve gf(x) = 10

7 marks
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a) 0 b) f-1(x) = x - 2 c) x = -4 or x = 3
7

f(x) = 1x + 2 for x ≠ −2   and   g(x) = 3x − 1

Find (fg)−1(x).

4 marks
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(1 - x)/(3x)
8

f(x) = x2 + 3 for x ≥ 0   and   g(x) = 2x − 1 for x ≥ 1.

Find (fg)−1(x).

4 marks
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(1 + sqrt(x - 3))/2
9

f(x) = 3x − 4   and   g(x) = x2 − 2

Solve f−1(x) = g(x).

4 marks
Show answer
x) 2 x) -5/3

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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