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Grade 7 · Higher · free GCSE maths worksheet

Domain and range of a function

Edexcel IGCSE

f: x maps to notation, values to exclude from the domain and the set of outputs.

10 original questions, 25 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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How to do it

  1. Domain = the inputs you are allowed to use. Range = the outputs you get
  2. Dividing? The bottom cannot be 0: solve bottom = 0 to find the value to exclude
  3. Square root? What is inside must be ≥ 0: solve that for the domain
  4. Finite domain: put the end values (and 0 for x²) into f to find the range
  5. Quadratic for all x: complete the square, the range is ≥ the number on the end

Watch out: If 0 is inside the domain, the smallest value of x² is 0, not the value at an end point.

Worked example

f(x) = 6x − 3

  1. State the value of x that must be excluded from the domain of f.

g : x ↦ 4x − 3   for   −2 ≤ x ≤ 3

  1. Find the range of g.

a) The bottom of a fraction cannot be 0

x - 3=0
+3x=3+3

x = 3 is excluded

b) Straight line, so the range comes from the two end points

g(-2)=4 × (-2) - 3 = -11
g(3)=4 × 3 - 3 = 9

-11 ≤ g(x) ≤ 9

The questions

1

The function f is defined by f : x ↦ 3x − 7

  1. Work out f(5).
  2. Work out f(−2).
2 marks
Show answer
a) 8 b) -13
2

f(x) = 5x − 6

State the value of x that cannot be included in the domain of f.

1 mark
Show answer
x = 6
3

The function f has domain {1, 2, 3, 4} and is defined by f : x ↦ x2 − 1

List the members of the range of f.

2 marks
Show answer
{0, 3, 8, 15}
4

h(x) = √(x − 4)   and   g(x) = 42x + 3

  1. State the largest possible domain of h.
  2. State the value of x that must be excluded from the domain of g.
3 marks
Show answer
a) x ≥ 4 b) x = -1.5 (accept x = -3/2)
5

f(x) = 3x − 2   for   −1 ≤ x ≤ 4

Find the range of f.

2 marks
Show answer
-5 ≤ f(x) ≤ 10 (accept -5 ≤ y ≤ 10)
6

The function f is defined by f : x ↦ 3x2 − 16

Find the two values of x that must be excluded from the domain of f.

2 marks
Show answer
x = 4 or x = -4
7

f(x) = x2   for   −3 ≤ x ≤ 2

Leon says that the range of f is 4 ≤ f(x) ≤ 9.

Leon is wrong. Find the correct range of f.

2 marks
Show answer
0 ≤ f(x) ≤ 9 (accept 0 ≤ y ≤ 9)
8

The function g is defined by g : x ↦ 2x   for   x > 0

  1. Work out g(0.4).
  2. State the range of g.
3 marks
Show answer
a) 5 b) g(x) > 0 (accept y > 0)
9

f(x) = 2x2 − 12x + 5   for all real values of x.

  1. Write f(x) in the form a(x − b)2 + c.
  2. Hence find the range of f.
4 marks
Show answer
a) 2(x - 3)2 - 13 b) f(x) ≥ -13 (accept y ≥ -13)
10

The function f is defined by f : x ↦ x2 + 3   with domain   x ≥ 0

  1. State the range of f.
  2. Write down the domain of the inverse function f−1.
  3. Write down the range of f−1.
4 marks
Show answer
a) f(x) ≥ 3 (accept y ≥ 3) b) x ≥ 3 c) f⁻¹(x) ≥ 0 (accept y ≥ 0)

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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