Topic 5 of 8
Functions as input–output rules
Step 10 · Pre-algebra topics
Think of a function as a rule that gives one output for each input, recognize linear functions from tables, rules and graphs, and read the story a graph tells.
Inputs and outputs
A function is a rule that gives exactly one output for each input. Think of a machine: put x in, and one value of y comes out.
| x | y = 3x − 1 |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 11 |
Example: Using a rule
The rule is "multiply by 4, then subtract 3". What is the output for 6?
- 4 × 6 = 24.
- 24 − 3 = 21.
Answer21
Example: Is it a function?
The pairs are (1, 5), (2, 7) and (1, 9). Is this a function?
- The input 1 has two outputs, 5 and 9.
AnswerNo: one input has two outputs
Linear functions
A linear function has a constant rate of change: each time x goes up by 1, y changes by the same amount. Its graph is a straight line, and it can be written y = mx + b.
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 4 | 7 | 10 | 13 |
Here y goes up 3 each time and starts at 4, so y = 3x + 4.
Example: Linear or not?
Is y = x² a linear function?
- x = 0, 1, 2, 3 gives y = 0, 1, 4, 9.
- The changes are 1, 3 and 5: not constant.
AnswerNo
Example: Comparing
Function A is y = 2x + 7. Function B goes up 3 for each 1 across and starts at 1. Which grows faster?
- A's rate of change is 2.
- B's rate of change is 3.
AnswerB
Reading graphs of functions
A graph tells a function's story. Read it from left to right: rising means the output is increasing, falling means it is decreasing, and flat means it stays the same.
Example: Reading a value
On the graph, what is the distance at time 2?
- Go up from 2 to the line, then across to the y-axis.
Answer2
Example: Describing
On the graph, when is the distance not changing?
- The line is flat from 4 to 6.
AnswerFrom 4 to 6
Try it yourself
Work each one on paper first, then open the answer to check.
The rule is y = 5x − 2. What is y when x = 3?
Show answer
13
The rule is "square it, then add 1". What is the output for 4?
Show answer
17
Is {(2, 3), (3, 3), (4, 3)} a function?
Show answer
Yes: each input has one output.
A table has x = 0, 1, 2 and y = 5, 8, 11. Write the rule.
Show answer
y = 3x + 5
Is y = 7 − 2x linear?
Show answer
Yes: y changes by −2 each time x goes up by 1.
Does a circle drawn on a grid show a function?
Show answer
No: a vertical line through the middle crosses it twice.