IGCSE Maths / Functions and function notation
Domain and range revision notes
Edexcel International GCSE Maths (4MA1), chapter 24: Functions and function notation, specification point 3.2c. Free lesson.
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The domain is the set of allowed inputs, and the range is the set of outputs a function can produce. Knowing which is which helps you read ranges from graphs and find them when the inputs are limited.
What domain and range mean
A function takes an input and applies a rule to produce an output. The domain describes what you are allowed to put in, while the range describes what can come out. A range must be written using the function output, such as f(x), rather than x. Changing the domain usually changes the range too. For example, f(x) ≥ 4 describes outputs. Writing an inequality with x describes inputs, so it gives a domain, not a range.
How to find the range from a rule or graph
For f(x) = x² + 4, the square of any number is at least 0. Adding 4 means the output is always at least 4, so the range is f(x) ≥ 4. On a graph, the range is the set of y-values the curve reaches. Look at the y-axis, not the x-axis. The curve has its lowest point at (0, 4), so its smallest output is 4. The inputs −2 and 2 both give an output of 8. Different inputs can give the same output. The curve reaches every y-value from 4 upwards, and none below 4.
How to find a range when the domain is limited
When the question gives a restricted set of inputs, test its two ends in the function rule. Then check whether the rule increases or decreases between them. If it changes steadily in one direction, the outputs include all the values between the endpoint outputs.
Check the output, not the input
A common error is to copy the domain straight into the range. The domain gives the allowed x-values, but the range gives the values produced by the rule. Use the function name in the range inequality and use the outputs you calculated. For h(x) = 7 − 2x, with 0 ≤ x ≤ 3, the rule decreases as x increases. The endpoint outputs are 7 and 1, so write the smaller first: 1 ≤ h(x) ≤ 7.