Functions, domain and range Cambridge IGCSE Additional Mathematics revision
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In plain words
A function is a machine: put a number in, and exactly one number comes out. The numbers you are allowed to put in are the domain. The numbers that can come out are the range.
6 things to know
- A function is a rule that gives exactly one output for every input.
- The domain is the set of inputs. The range (or image set) is the set of outputs.
- A function is one–one if every output comes from exactly one input. It is many–one if some output comes from more than one input, as with x², where 3 and −3 both give 9.
- Notation: f(x) = 2x + 1 and f : x ↦ 2x + 1 mean the same thing.
- To find a range, sketch the graph over the domain and read off the lowest and highest values of y. For a quadratic, find the turning point first.
- A rule that gives two outputs for one input, such as ±√x, is not a function.
Worked example
Find the range of f(x) = x² + 3 for all real x, and say whether f is one–one.
- x² is never negative, and it is 0 when x = 0. So the smallest value of f(x) is 3.
- There is no largest value. The range is f(x) ≥ 3.
- f(2) = 7 and f(−2) = 7. Two inputs give the same output, so f is many–one, not one–one.
Worked example
Find the range of f(x) = 2x − 1 for 0 ≤ x ≤ 5.
- The graph is a straight line going up, so the smallest value is at x = 0 and the largest at x = 5.
- f(0) = −1 and f(5) = 9. The range is −1 ≤ f(x) ≤ 9.
Tips and tricks
- Write a range in terms of f(x) or y, and a domain in terms of x. Writing "x ≥ 3" for a range loses the mark.
- To show a function is not one–one, give two inputs with the same output.
It lands in your notebook with its questions as flashcards.
Functions, domain and range: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the domain of a function?
The range is the set of outputs.
What is the range of f(x) = 5 − x² for all real x?
x² is at least 0, so 5 − x² is at most 5.
Which function is one–one for all real x?
A straight line that is not horizontal gives each output once only.
f(x) = x² − 4. What is f(−3)?
(−3)² = 9, and 9 − 4 = 5.
What is the range of f(x) = eˣ for all real x?
eˣ is always positive, and never reaches 0.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 6 marks. Write your answers on paper, then check them.
Functions, domain and range
Cambridge IGCSE Additional Mathematics 0606 · 6 marks · papermunch.org
Name ______________________________ Date ______________
Find the range of f(x) = 3x + 2 for −1 ≤ x ≤ 4.[2]
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−1 ≤ f(x) ≤ 14. f(−1) = −1 and f(4) = 14.
Explain why f(x) = x², for all real x, is not a one–one function.[2]
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Two different inputs give the same output. For example, f(3) = 9 and f(−3) = 9.
State the range of f(x) = (x − 2)² + 5 for all real x.[2]
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f(x) ≥ 5. The square is at least 0, so the smallest value is 5.
Answers: Functions, domain and range
- 1. −1 ≤ f(x) ≤ 14. f(−1) = −1 and f(4) = 14.
- 2. Two different inputs give the same output. For example, f(3) = 9 and f(−3) = 9.
- 3. f(x) ≥ 5. The square is at least 0, so the smallest value is 5.