Domain and Range for IGCSE Maths
Identifying the set of allowed inputs and possible outputs of a function. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding domain
What You Need to Know
Identifying the set of allowed inputs and possible outputs of a function. This subtopic is part of Functions in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding domain and range is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Domain and Range
The domain of a function is the set of valid input values (x-values). The range is the set of corresponding output values (y-values). For IGCSE Extended, you must determine restricted domains (e.g. x ≠ 2 for 1/(x−2), x ≥ 0 for √x) and state the corresponding range. Domains and ranges also govern when an inverse function exists: a function must be one-to-one on its domain for an inverse to be defined.
Step-by-Step Method
- 1
Identify restrictions on x
Ask: what values of x cause problems? Division by zero (denominator = 0 is excluded), square root of negative (radicand ≥ 0 required), log of non-positive (argument > 0 required).
- 2
State the domain
Write the domain as an inequality or set: e.g. x > 2, x ≥ 0, x ≠ −1.
- 3
Find the range by analysing the function's behaviour
For a quadratic ax²+bx+c, complete the square to find the minimum (or maximum). The range is y ≥ min (or y ≤ max) for a > 0 (or a < 0).
- 4
For composite/inverse, track domain and range through each step
The domain of fg is the set of x-values where g(x) is defined AND lies in the domain of f.
- 5
Sketch the graph for confirmation
A sketch of y = f(x) makes the range visually clear — the y-values the graph takes on.
Worked Example
Question
f(x) = √(3x − 6). (a) Write down the domain of f. (b) Find the range of f.
Solution
Part (a): The expression under the square root must be ≥ 0. 3x − 6 ≥ 0 3x ≥ 6 x ≥ 2 Domain: x ≥ 2 Part (b): When x = 2: f(2) = √0 = 0 (minimum output). As x → ∞, f(x) → ∞. Range: f(x) ≥ 0
Exam Tips for Domain and Range
- Domain is about x-values (inputs); Range is about y-values (outputs) — mix these up and you lose marks.
- For square roots, set the radicand ≥ 0 and solve for x to find the domain.
- A quadratic's range is found by completing the square — the vertex gives the minimum or maximum y-value.
- When given a restricted domain like −2 ≤ x ≤ 5, evaluate f at both endpoints and the vertex to determine the full range.
Practice Questions
Q1: g(x) = 1/(x − 3). State the domain and describe the behaviour as x → 3.
Show hint
Domain: x ≠ 3 (or x ∈ ℝ, x ≠ 3). As x → 3 from above, g → +∞; from below, g → −∞.
Q2: h(x) = x² − 4x + 1. Find the range, given that x can be any real number.
Show hint
Complete the square: h(x) = (x−2)²−3. Minimum value is −3. Range: h ≥ −3.
Q3: f(x) = 2x + 5 for −3 ≤ x ≤ 4. Find the range of f.
Show hint
f(−3) = −1 (minimum), f(4) = 13 (maximum). Range: −1 ≤ f ≤ 13.
Frequently Asked Questions
What is domain and range in IGCSE Maths?
Identifying the set of allowed inputs and possible outputs of a function.
Is domain and range in the Core or Extended syllabus?
Domain and Range is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise domain and range effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on domain and range rather than trying to cover everything at once.
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