What Makes a Function?

Algebra 1 · Axiom Academy

The one rule that defines every function — and the Vertical Line Test that lets you spot one at a glance. 1. The Core Rule: One Input, One Output A function connects a set of inputs (the domain ) to a set of outputs (the range ). It follows one strict, non-negotiable rule. Watch the arrows: each input on the left must point to exactly one output on the right. 2. The Vertical Line Test: Passing On a graph, a vertical line is the set of all points sharing one input x . So if a vertical line ever meets the curve more than once , that single input has more than one output — and the rule fails. Watch a vertical line sweep across the parabola y = x^2 . Now apply the same sweep to a circle, x^2 + y^2 = r^2 . A circle is a perfectly good relation — but watch the hit counter as the line crosses the middle. Every vertical line: 1 hit. Passes — a function. Lines through the middle: 2 hits. Fails — not a function. 4. Try the Test on Four Graphs Here is the real payoff: one quick visual check classifies any graph. Watch the same vertical sweep run across four curves at once. The two that a line can ever hit twice are not functions; the two it only ever hits once are . You can now state what makes a rule a function and spot one on a graph in a single glance. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Algebra 1 course.