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0606/11 Oct/Nov 2025, Question 4: Functions: One-One, Inverses and Restricted Domains

Cambridge IGCSE Additional Mathematics 0606 · Paper 11 (Non-calculator) · Non-calculator · 10 marks

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The question, in short

Questions on functions: explain why a circle graph is not a function, tick properties of four graphs, and work with f(x)=sin x and g(x)=√(x−1/2) including gf and g².

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

State why the circle graph fails the definition of a function, referring to its x-values or vertical line test.

  1. 1A function gives exactly one output for each input
  2. 2A vertical line through the circle meets it twice, so one x has two y values

AnswerIt is a many-many mapping: a vertical line meets the graph more than once

Part (b) · 4 marks

For each of the four graphs, tick whether it is one-one, many-one, and its own inverse.

  1. 1Graph 1: a horizontal line can meet it more than once → many-one
  2. 2Graph 2: y = 4 − x on 0 ≤ x ≤ 4 is one-one and symmetric in y = x → own inverse
  3. 3Graph 3: horizontal lines meet it several times → many-one
  4. 4Graph 4: strictly increasing → one-one, not symmetric in y = x

AnswerGraph 1: many-one; Graph 2: one-one and its own inverse; Graph 3: many-one; Graph 4: one-one

Part (ci) · 2 marks

Find the largest value of a so that gf exists, using sin 30° = 1/2.

  1. 1g needs its input ≥ 1/2, so sin x ≥ 1/2
  2. 2sin x ≥ 1/2 for 30° ≤ x ≤ 150°
  3. 3Largest domain: a = 150

Answera = 150

Part (cii) · 2 marks

For the domain from part (i), find the range of the composite function gf.

  1. 1For 30° ≤ x ≤ 150°, 1/2 ≤ sin x ≤ 1
  2. 2gf(x) = √(sin x − 1/2)
  3. 3Least value √0 = 0, greatest value √(1/2) = 1/√2

Answer0 ≤ gf(x) ≤ 1/√2

Part (ciii) · 1 mark

Decide whether the function g² exists, justifying briefly.

  1. 1Range of g is g(x) ≥ 0
  2. 2Domain of g is x ≥ 1/2
  3. 3Values 0 ≤ g(x) < 1/2 cannot be put into g, so g² does not exist

AnswerNo: the range of g (g(x) ≥ 0) is not contained in the domain of g (x ≥ 1/2)

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