Functions and Relations
A function gives exactly one output for each input.
About this topic
A relation is any set of ordered pairs linking inputs to outputs. A function is a special relation in which each input has exactly one output. You can show functions as tables, mappings, equations or graphs.
The vertical line test checks a graph: if any vertical line crosses the graph more than once, then some input has two outputs and it is not a function. A circle fails the test, while a line or a parabola opening up passes. A table with the pair (2, 5) and also (2, 7) is not a function, but (2, 5) and (3, 5) is fine, since different inputs may share an output.
Functions are a central organizing idea in mathematics and science. A vending machine, a recipe or a temperature converter all behave like functions: given the same input you always get the same output. Programmers write functions in code for exactly this reason.
Students often mix up the rule. The restriction is on inputs, not outputs: one input may never give two different results. The relation x = y² is not a function of x, because x = 4 gives both y = 2 and y = −2.
Key ideas
- Vertical line test checks functions
- Inputs are the domain, outputs the range
- Mapping diagrams show pairings
Quick check
Use cases: where Functions and Relations helps
- Homework: review the key idea behind Functions and Relations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Functions and Relations without notes.
- Visual learning: play with the interactive lab to see how Functions and Relations changes when you adjust its inputs.
- Teaching a friend: explain Functions and Relations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Functions and Relations?
A function gives exactly one output for each input.
What will I learn in Functions and Relations?
Vertical line test checks functions. Inputs are the domain, outputs the range. Mapping diagrams show pairings.
Is Functions and Relations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Functions and Relations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Functions and Relations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.