A function is a rule that gives exactly one output for each input.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Think of a machine
Put a number in, one number comes out. The same input always gives the same output.
What breaks it
If one input could give two different outputs, the rule is not a function.
An example that fails
"A number and its square roots" gives 4 both 2 and −2, so it is not a function.
One output for every input
A function is a rule assigning exactly one output to each input. The "exactly one" is the whole definition. A rule that gives two different outputs for the same input is a relation but not a function.
The vertical line test
On a graph, if any vertical line crosses the curve more than once, the same input has two outputs and the graph is not a function. A circle fails; a straight line passes. The test is a visual restatement of the definition.
Function notation
Writing f(x) = 3x + 1 names the rule f, and f(2) = 7 means the output at input 2. The brackets do not mean multiplication. The notation lets you refer to and compare several rules without confusion.
Why functions matter
Nearly everything that depends on something else is a function: cost on quantity, distance on time, population on year. Functions are the language for describing dependence, and almost all later mathematics is built in it.
Step 2: Try It Yourself
Tap and try it out.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Leo Test Two Rules
Leo checks "double it" and "a number and the numbers that square to it".
- Step 1
Doubling 5 gives 10, and it always gives 10.
Step 4: Your Turn
Practice makes it stick.
The Rule
Problem 1 of 2
The rule is "add 7". What comes out when 5 goes in?
The Test
Problem 2 of 2
A rule sends 4 to both 2 and −2. Is it a function?
Function or Not
1 of 4
Rule "multiply by 3". What comes out when 6 goes in?
2 of 4
Is "double it" a function?
3 of 4
Rule "subtract 4". What comes out when 10 goes in?
4 of 4
Can one input give two outputs in a function?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Rule "multiply by 4 then add 1". What comes out when 5 goes in?
Question 2 of 2
What makes a rule a function?
What You Learned
- A function gives exactly one output for each input.
- The same input always gives the same output.
- Two outputs from one input means it is not a function.