What Is a Function in Math?
A function is a rule that takes an input and produces exactly one output.
Think of it as a machine:
- You put a number in (input)
- The machine does something to it (the rule)
- One number comes out (output)
Example
The function "double and add 1":
- Input 3 → output 7 (3 × 2 + 1 = 7)
- Input 5 → output 11 (5 × 2 + 1 = 11)
- Input 0 → output 1 (0 × 2 + 1 = 1)
Written mathematically: f(x) = 2x + 1
"f(x)" is read as "f of x." It means "the output of function f when the input is x."
The one key rule
Every input must produce exactly one output. If you put 3 into the machine, it cannot sometimes give you 7 and sometimes give you 9. One input → one output. Always.
However, different inputs can produce the same output. f(x) = x² gives: f(3) = 9 and f(-3) = 9. Two different inputs, same output — that is allowed.
Functions as tables
| Input (x) | Output f(x) = 2x + 1 |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
Functions as graphs
Plot the input-output pairs as (x, y) coordinates on the coordinate plane. The function f(x) = 2x + 1 produces a straight line.
Where functions appear
Functions are the language of math from Grade 8 onward:
- Linear equations: y = mx + b is a function
- Area formulas: A = πr² is a function of radius
- Probability: probability distributions are functions
- Every formula your child has ever used (A = l × w, V = l × w × h) is a function
Related concepts
- Variables and equations: functions use variables
- Coordinate plane: functions are graphed on the coordinate plane
- Linear equations: the simplest type of function