A relation is a function when every input has exactly one output.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The vertical line test
If any vertical line crosses the graph twice, one input has two outputs and it is not a function.
Outputs may repeat
Two different inputs sharing an output is fine. A constant function does it everywhere.
In a table
Look for a repeated input with different outputs. That is the only thing that disqualifies it.
One output per input
A relation is a function when each input has exactly one output. Two inputs sharing an output is fine; one input with two outputs is not. The asymmetry is the whole definition and it is easy to state backwards.
The vertical line test
If any vertical line meets a graph more than once, some input has two outputs, so the graph is not a function. A circle fails, a parabola opening upwards passes, and a sideways parabola fails.
Checking a table or a set of pairs
Look for a repeated input with different outputs. {(1,2), (2,3), (1,5)} is not a function because 1 appears twice with different partners. {(1,2), (2,2)} is fine — repeated outputs are permitted.
Why the restriction matters
Functions can be inverted, composed and differentiated in ways general relations cannot. Almost every tool in later mathematics assumes single-valuedness, which is why the definition is enforced so early and so strictly.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Dev Test Two Tables
Table A has inputs 1, 2, 3. Table B has inputs 1, 1, 2.
- Step 1
In A every input appears once, so it is a function.
Step 4: Your Turn
Practice makes it stick.
The Repeat
Problem 1 of 2
Input 4 maps to both 7 and 9. Is it a function? 1 for yes, 0 for no.
The Shared Output
Problem 2 of 2
Inputs 2 and 5 both map to 8. Is it a function? 1 for yes, 0 for no.
Function or Not
1 of 8
A vertical line crosses a graph twice. Function? 1 for yes, 0 for no.
2 of 8
A horizontal line crosses a graph twice. Function? 1 for yes, 0 for no.
3 of 8
Is a circle a function? 1 for yes, 0 for no.
4 of 8
Is y = 5 a function? 1 for yes, 0 for no.
5 of 8
Sort each relation by whether it is a function.
Tap something to move it.
- Empty
- Empty
6 of 8
Is x = 2 a function? 1 for yes, 0 for no.
7 of 8
Inputs 1, 2, 3 map to 4, 4, 4. Function? 1 for yes, 0 for no.
8 of 8
How many outputs may one input have in a function?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Input 3 maps to both 1 and 8. Function? 1 for yes, 0 for no.
What You Learned
- A function gives exactly one output for every input.
- The vertical line test catches inputs with two outputs.
- Different inputs may share an output; that breaks nothing.