---
title: "What Is the Range of a Function?"
url: https://www.learningloftinstitute.com/glossary/range-of-a-function
source: Learning Loft Institute
language: en
updated: 2026-08-21
section: "Glossary › What Is the Range of a Function?"
---
# What Is the Range of a Function?
> **Answer.** The range of a function is the set of values it actually outputs. For f(x) = x² + 1 the range is f(x) ≥ 1, because a square is never negative and the smallest input contribution is zero. For f(x) = 1/x it is every real number except 0, since no input produces zero.
## Key facts
| | |
| --- | --- |
| Definition | The set of output values a function actually produces once every input in its domain has been used. |
| Where students meet it | Grade 9 or 10 mathematics, always in the same lesson as domain, and on IGCSE Extended papers where both are asked for in one question. |
## Read it off the vertical axis
Sketch the graph and the range is the set of heights the curve occupies. Domain is a question about how far the picture spreads sideways; range is a question about how far it spreads up and down. Asking which y values are used is usually faster than any algebra.
y = x² + 1 sits on top of its lowest point, (0, 1), and climbs from there in both directions, so every height from 1 upwards is used and nothing below. y = 1/x is in two separate pieces, one above the axis and one below, and between them they use every height except the one the curve never reaches.
For a curve without a lowest point the answer is simply all real numbers. Any non-horizontal straight line, such as y = 3x − 4, has that range: continue far enough in either direction and every height is eventually reached.
## What a mapping diagram shows, and what it does not
On a mapping diagram the inputs are drawn on the left and the outputs on the right, with arrows between them. The left-hand column is the domain. The range is not the right-hand column — it is only the part of it that has an arrow pointing at it.
That distinction has a name. The set you draw on the right is the codomain, the values the function is permitted to produce; the range is the values it actually produces. For f(x) = x² you might write all real numbers on the right, but no arrow ever lands on a negative, so the range is f(x) ≥ 0. Exam questions want the values reached.
## Change the domain and the range moves
The two are tied together, which is why questions hand you one and ask for the other. f(x) = x² over every real number has range f(x) ≥ 0. The same rule restricted to 1 ≤ x ≤ 3 has range 1 ≤ f(x) ≤ 9, because the outputs now stop at both ends.
For a restricted domain, the safe method is to evaluate the function at both endpoints and then check whether a turning point sits between them. For f(x) = x² − 6x + 5 on 0 ≤ x ≤ 5 the endpoints give 5 and 0, but the minimum at x = 3 gives −4, so the range runs from −4 up to 5. Skipping the turning point is how the wrong answer gets written.
## Questions
### How do you find the range without drawing the graph?
Ask what the rule can and cannot produce. A squared term is never negative, so anything of the form x² + k has range at least k. A fraction with 1 on top never gives zero. Completing the square finds the lowest or highest value of a quadratic exactly, which is usually all the range question needs.
### Is the range the same as the codomain?
No. The codomain is the set of values a function is allowed to output; the range is the set it actually outputs. For f(x) = x² the codomain might be all real numbers, but the range is only f(x) ≥ 0. School questions almost always want the range, so the values actually reached.
### What is the range of a linear function like y = 3x − 4?
All real numbers. A straight line with a non-zero gradient keeps rising and falling without limit, so every height is used somewhere along it. The exception is a horizontal line such as y = 2, whose range is the single value 2, since that is the only output it ever gives.
### Should the answer use y or f(x)?
Either is accepted, but stay consistent with how the question was written. If it defines f(x) = x² + 1, answer f(x) ≥ 1. If it gives y = x² + 1, answer y ≥ 1. Writing x ≥ 1 answers a domain question by mistake and is the most common way the mark is lost.
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