What Makes a Rule a Function

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Idea

A proposed rule must be defined for every allowed input, must produce outputs in its codomain, and must assign only one output per input. Graphically, uniqueness is the vertical-line test.

Method

A vertical line $x=a$ may meet the graph at at most one point. A formula that fails at an input can define a function on a smaller domain, but cannot silently retain the forbidden input.

Worked example

The circle $x^2+y^2=1$ fails the vertical-line test: at $x=0$ it has $y=1$ and $y=-1$ . Its upper half $y=\sqrt{1-x^2}$ is a function on $[-1,1]$ . The rule $1/x$ is a function on $\mathbb R\setminus\{0\}$ , but not on all of $\mathbb R$ .

Common mistake

The horizontal-line test checks one-to-one behavior, not whether the relation is a function in the first place.

Check your understanding

Is $y=|x|$ a function for real $x$ ?

Show answer

Yes. Every real input has one nonnegative absolute value.

Explore

Domain and range

Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.

Function

y = x² · Domain: all real x · Range: y ≥ 0

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.

Does the full circle x²+y²=1 define y as a function of x?

Yes: it is a smooth curve.

Yes: x is restricted to [−1,1].

No: at x=0 there are two y values.

Hint 1 · Find a starting point

Use the vertical-line test.

Hint 2 · Take the next step

At x=0, solve y²=1.

Show the reasoning

Answer: No: at x=0 there are two y values.

Both y=1 and y=−1 occur for one input, so the full circle is not such a function.

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

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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