Defining a Function

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Idea

A function assigns exactly one output to each input in its domain. Think of the input as a question and the output as its unambiguous answer. Different inputs may share an output; one input cannot have two outputs.

Visual guide

VISUAL GUIDEEach input gets exactly one output

Squaring on three inputs

Two different inputs may share an output. This is still a function because each input has one outgoing arrow. A rule with two different outputs for a single input would fail the definition.

Method

The notation $f:A\to B$ specifies the domain $A$ and codomain $B$ . The rule $f(x)$ tells how to assign outputs. The actual outputs form the range, which is a subset of $B$ .

Worked example

Let $f:\mathbb R\to\mathbb R$ have rule $f(x)=x^2$ . Then $f(-3)=9$ and $f(3)=9$ . These equal outputs do not violate the function definition: each input still has only one square. The equation $y^2=x$ describes a relation with two possible $y$ values for positive $x$ , unless a branch is selected.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which assignment fails to define a function?

Every input has exactly one output.

The same input 2 has outputs 3 and 5.

Two different inputs share output 3.

Hint 1 · Find a starting point

A function’s rule assigns one output to each allowed input.

Hint 2 · Take the next step

Repeating an output is allowed; competing outputs for one input are not.

Show the reasoning

Answer: The same input 2 has outputs 3 and 5.

Input 2 cannot simultaneously be assigned two different outputs.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

$f(x)$ is the output, not multiplication of $f$ by $x$ . The input symbol is a placeholder: $f(t)=t^2$ describes the same rule.

Check your understanding

Does $\{(1,2),(2,2),(1,3)\}$ define a function of the first coordinate?

Show answer

No. Input $1$ has outputs $2$ and $3$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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