In dependent and independent variables one quantity depended on another. A function makes that idea precise: it is a rule you can count on.
Why it works
A rule is only useful if it is predictable. If you put 3 into a machine and sometimes got 5 and sometimes 7, you could never say what 3 gives. So the one requirement is that each input has a single output. Two different inputs may give the same output, just as two students can share a birthday; that does not make anything unpredictable.
Key idea
A function assigns to each input exactly one output. To test a list, table or diagram, look for an input that appears with two different outputs. If there is none, it is a function.
Worked example: A list of pairs
Is $\{(-2, 3), (0, 1), (4, 3), (5, 0)\}$ a function?
- The inputs are $-2$, 0, 4 and 5: each appears once.
- Two inputs share the output 3, which is fine.
Answer: Yes, it is a function.
Worked example: A table
Is this a function? $$\begin{array}{c|cccc} x & 1 & 2 & 2 & 3 \\ \hline y & 6 & 8 & 9 & 10 \end{array}$$
- The input 2 appears twice, with outputs 8 and 9.
- One input with two different outputs breaks the rule.
Answer: No, it is not a function.
Worked example: Points on a graph
The points $(1, 2)$, $(1, 5)$ and $(3, 4)$ are plotted. Do they show a function?
- $(1, 2)$ and $(1, 5)$ have the same input, 1, but different outputs.
- On a graph, that shows up as two points on the same vertical line.
Answer: No.
Common mistake
Thinking a function can’t have two inputs with the same output. Repeated outputs are fine. Only a repeated input with different outputs breaks the rule.