Relations vs Functions
A function gives exactly one output per input (vertical line test).
Key idea
A function gives exactly one output per input (vertical line test).
In depth
A relation is any set of pairs. A function is a relation where every input has exactly one output. Graphically, the vertical line test: if any vertical line hits the graph more than once, it is not a function.
Worked example
{(1,2), (1,3)} is not a function. A circle fails the vertical line test.
Real-world uses
- Checking whether a table of data assigns one output per input before modelling it
Common mistake
Thinking each output must come from only one input. Different inputs can share an output.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Relations vs Functions?
A function gives exactly one output per input (vertical line test). It is a topic at the intermediate level in Algebra.
Where is Relations vs Functions used in real life?
Real-world examples: Checking whether a table of data assigns one output per input before modelling it.
What should I learn before Relations vs Functions?
Make sure you are comfortable with Domain and Range, the previous topic in Algebra.
What is a common mistake with Relations vs Functions?
Thinking each output must come from only one input. Different inputs can share an output.
How can I practise Relations vs Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Relations vs Functions?
Move on to Systems of Equations: Graphing next.
Related intermediate topics in Algebra
- Multi-Step Equations – Distribute, combine like terms, move variables to one side.
- Variables on Both Sides – Collect variable terms on one side first.
- Literal Equations – Rearrange formulas to solve for a chosen variable.
- Identity and No-Solution Equations – Some equations are always true or never true.
- Linear Inequalities – Solve like equations but reverse the sign when multiplying by a negative.
- Compound Inequalities – Combine two inequalities using “and” or “or”.