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Grade 7 · Higher · free GCSE maths worksheet

Composite and inverse functions

fg(x) and f^-1(x).

13 original questions, 58 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) = 3x − 4    g(x) = x2

  1. Find f(5)

  2. Find gf(2)

3 marks
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a) 11 b) 4
2

f(x) = 2x + 5

  1. Find f−1(x)

  2. Find f−1(11)

3 marks
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a) f-1(x) = (x - 5)/2 b) 3
3

f(x) = x + 23    g(x) = x2 − 1

  1. Find fg(3)

  2. Find f−1(x)

4 marks
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a) 10/3 b) f-1(x) = 3x - 2
4

f(x) = 2x − 1    g(x) = x2 + 3

  1. Show that gf(x) = 4x2 − 4x + 4

  2. Hence find gf(3)

4 marks
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a) gf(x) = (2x - 1)2 + 3 = 4x2 - 4x + 4 b) 28
5

f(x) = x2 + 2x

Express f(x + 3) in the form x2 + bx + c where b and c are integers.

3 marks
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x2 + 8x + 15
6

f(x) = 3x + 2    g(x) = x2 − 4

  1. Find f−1(x)

  2. Solve gf(x) = 0

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

f(x) = 4x − 1    g(x) = x2 + 2

  1. Find fg(−3)

  2. Find gf(x), giving your answer in the form ax2 + bx + c

  3. Find f−1(x)

5 marks
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a) 43 b) 16x2 - 8x + 3 c) f-1(x) = (x + 1)/4
8

f(x) = x2    g(x) = x + 2

  1. Find fg(x), giving your answer in the form x2 + bx + c

  2. Solve fg(x) = gf(x)

5 marks
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a) x2 + 4x + 4 b) x = -1/2
9

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
10

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
11

f(x) = 1x − 4   and   g(x) = √(x + 3)

  1. State the value of x that must be excluded from the domain of f.
  2. State the largest possible domain of g.
  3. State the range of g.
  4. fg(x) is not defined for one value of x in the domain of g. Find this value.
4 marks
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a) x = 4 b) x >= -3 c) g(x) >= 0 d) x = 13
12

f(x) = 3x + 10   and   g(x) = sin x°

  1. Work out fg(30).
  2. Work out gf(20), correct to 3 decimal places.
  3. Ahmed solves gf(x) = 0.5 and writes “3x + 10 = 30, so x = 203 is the only solution.” Show that Ahmed is wrong by finding another solution between 40 and 50.
5 marks
Show answer
a) 11.5 b) 0.940 c) x = 140/3 (46.67)
13

f(x) = x2 − 6x + 11 for all real x.

  1. Write f(x) in the form (x − a)2 + b.
  2. Hence state the range of f.
  3. Explain why f does not have an inverse function for all real x.
4 marks
Show answer
a) (x - 3)2 + 2 b) f(x) >= 2 c) f is many-to-one, e.g. f(2) = f(4) = 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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