A function gives each input exactly one output. If any vertical line crosses the graph twice, some input has two outputs — and that settles it on sight.
Read the graph
Is the graphed relation a function?
The dashed line at x = 1.5 meets the circle at two heights, so the input 1.5 has two outputs — not a function. Smoothness (A) and straightness (D) are irrelevant: parabolas are curved and pass easily. C tests the wrong direction — functions restrict what each x maps to.
A function is a machine: feed it an input x, get back exactly one output y. Two different inputs may share an output — that is allowed — but one input can never return two answers. The graph version of that promise: no x-value may sit under two points of the curve.
Mentally slide a vertical line from left edge to right edge.
every position: 0 or 1 crossing
it is a function
any position: 2+ crossings
not a function — one failure is enough
On the circle, x = 1.5 crosses at two heights — one bad line, test over. You never need to check the rest.
A vertical line itself, such as x = −1, is the worst case: the input −1 has infinitely many outputs, so it is not a function. A horizontal line such as y = 3 passes — every input gets the one output 3.
Where does a vertical line meet a circle only once? At its leftmost and rightmost points. For centre (6, 1) and radius 2, those are x = 6 − 2 = 4 and x = 6 + 2 = 8; every x strictly between them meets the circle twice, and every x outside misses it.
function ✓
function ✓ — curved is fine
not a function ✗
not a function ✗
| Test | Question it answers |
|---|---|
| vertical line | is it a function at all? |
| horizontal line | is the function one-to-one (does an inverse exist)? |
y = x² passes the vertical test — it is a function — but fails the horizontal one, since y = 4 comes from both 2 and −2. Exams love offering "fails the horizontal line test" as a reason something is not a function. It never is.
At a jump, the vertical line through the jump decides everything. Only filled dots are points of the graph; an open circle is a hole.
| At x = 2 the graph shows | Points on x = 2 | Function? |
|---|---|---|
| open circle at (2, 3), filled dot at (2, 0) | one: (2, 0) | yes ✓ |
| filled dot at (2, 3), filled dot at (2, 0) | two | no ✗ |
| open circles at both (2, 3) and (2, 0) | none — 2 is not in the domain | yes ✓ |
| a lone extra dot at (2, 5) above a curve through (2, 1) | two | no ✗ |
A jump, separate pieces or a stray point are all allowed in themselves. The only question is whether some x ends up with two filled points.
Worked: the sideways parabola x = y² − 4 fails the test. Which part can we keep to make it a function?
y² = x + 4 → y = ±√x + 4
solve for y: the ± is the two halves
keep y ≥ 0: y = √x + 4
the top half — one height per x
or keep y ≤ 0: y = −√x + 4
the bottom half works too
Restricting x (say "x ≥ 0") still leaves a top point and a bottom point over each x, so it fails. The cut must separate the two outputs — a condition on y. The same idea joins two rules into one function: their x-ranges must not overlap, not even at a single shared endpoint.
Besides "which graph is a function", expect jumps with open and filled dots, a stray point, half-circles, a restriction that rescues a relation, and one-to-one questions. Hunt for the single worst vertical line — at a jump, a stray dot or the widest part of a curve — and count filled points only.
Read the graph
The graph shows x = y². Which statement is true?
The dashed line at x = 2 crosses the curve at two heights, √2 and −√2 — one input, two outputs, so it is not a function. B assumes all parabolas pass; only the upright ones do. D cites the wrong test — the horizontal test is about inverses, never about whether something is a function.
