Functions
A function is a rule that gives exactly one output for every input value.
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Play with the controls to see the idea in action.
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A function is a rule that gives exactly one output for every input value.
- Written f(x) for input x
- The domain is the set of allowed inputs
- The range is the set of outputs
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Functions?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Functions. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Functions in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Functions gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Functions in detail
A function is a rule that takes an input and gives exactly one output, written f(x). For f(x) = 2x + 3, the input 4 gives f(4) = 11. The set of allowed inputs is the domain and the set of outputs is the range.
Functions can be shown as formulae, tables, graphs or mapping diagrams, and a graph passes the vertical line test if each x-value gives only one y-value. Thinking in functions gives a general way to describe relationships such as cost against quantity, and it leads to composite and inverse functions.
How the interactive helps you understand Functions
Move the sliders to change the values in the equation. The curve, the moving point and, where they exist, the roots (the points where the curve crosses the x-axis) update straight away.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Functions.
Real-life uses and examples
- Spot Functions in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Functions by estimating first, then working it out step by step.
- Use Functions when comparing prices, discounts or recipe quantities in everyday life.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Functions.
- Memorising the steps for Functions without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Functions
What is Functions?
A function is a rule that gives exactly one output for every input value.
What are the most important points about Functions?
The main points are: Written f(x) for input x; The domain is the set of allowed inputs; The range is the set of outputs.
Why is Functions useful?
Understanding Functions gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Functions?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Functions?
Related topics in Mathematics include Angles in Parallel Lines, Quadrilaterals, Similarity.