Worked examples — Vertical line test for functions

This page is the drill hall for the vertical line test. The parent note told you the rule: a graph is a function when no vertical line hits it more than once. Here we hunt down every kind of graph that rule can be aimed at, so that when an exam throws a weird curve at you, you have already seen its cousin.

Before we start, one word we lean on constantly: a vertical line is a line of the form $x=a$ (a fixed number $a$, with $y$ free to be anything). Picture holding a ruler perfectly upright and sliding it left–right across the page. At each stop, count how many times it stabs the graph. That count is the whole game.

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The scenario matrix

Every problem this topic can hand you falls into one of these case classes. The examples below are labelled with the cell they cover, and together they fill the whole table.

Cell	Case class	Example that covers it
A	Explicit $y=f(x)$ — always passes	Ex 1
B	Even power on $y$ (e.g. $y^2=\dots$) — two branches	Ex 2
C	Circle / ellipse — closed loop, top & bottom	Ex 3
D	Vertical line itself $x=c$ — degenerate, infinitely many hits	Ex 4
E	Horizontal line $y=c$ — passes, only one $y$ ever	Ex 4
F	Piecewise with a filled/open endpoint clash	Ex 5
G	Function with a gap (undefined at a point)	Ex 6
H	Oscillating graph (looks scary, still passes)	Ex 7
I	Real-world word problem (time → temperature)	Ex 8
J	Exam twist: same equation, two questions	Ex 9

The key axes of the matrix: how many $y$-values a chosen $x$ produces (0, 1, or many), and whether the graph is written as $y=\dots$ or as a mixed relation. If you can classify a graph into one of A–J, you can answer instantly.

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Example 1 — Cell A: a clean $y=f(x)$

Forecast: Take a guess before reading on — does a straight slanted line ever get hit twice by an upright ruler?

1. Pick any $x=a$ and solve for $y$. At $x=a$, $y=2a-1$. Why this step? The vertical line $x=a$ collects every point whose first coordinate is $a$. To count intersections, I ask: how many $y$ satisfy the equation once $x$ is nailed to $a$?
2. Count the answers. The expression $2a-1$ is a single number. One $a$ in, exactly one $y$ out. Why this step? "Exactly one output per input" is the literal definition of a function — I'm checking it directly.
3. Every $a$ behaves the same. Nothing about $2a-1$ can produce two numbers.

Verify: Try $a=3$: $y=2(3)-1=5$. Just the point $(3,5)$, one hit. ✓ Passes — it is a function.

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Example 2 — Cell B: an even power hiding on the $y$ side

Forecast: The $x$ side is innocent, but look where the square sits. Guess: pass or fail?

1. Fix $x=a$ and solve for $y$. $y^2=a⟹y=+\sqrt{a}$ or $y=-\sqrt{a}$. Why this step? The square root of a positive number has two answers, one positive and one negative. That "$\pm$" is exactly two $y$-values for one $x$.
2. Test a concrete value. At $x=4$: $y^2=4⟹y=2$ and $y=-2$. Two points: $(4,2)$ and $(4,-2)$. Why this step? One counterexample is enough to fail the test — I only need a single vertical line that hits twice.
3. Notice the special columns. At $x=0$, $y^2=0$ gives only $y=0$ (one hit). At $x=-3$, $y^2=-3$ has no real solution (zero hits). But those don't save it — the test fails the moment any line hits twice.

Verify: $2^2=4$ and $(-2{)}^2=4$ — both genuinely lie on the curve. ✗ Fails — not a function.

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Example 3 — Cell C: the circle (closed loop)

Forecast: A circle is a loop with no gaps. Where would a vertical ruler cut it — how many times?

1. Solve for $y$ at a fixed $x=a$. $y^2=25-a^2⟹y=\pm \sqrt{25-a^2}$. Why this step? Same manoeuvre as Ex 2 — isolate $y$ so I can count how many values come out.
2. Interior columns give two hits. Pick $x=3$: $y=\pm \sqrt{25-9}=\pm \sqrt{16}=\pm 4$. Points $(3,4)$ and $(3,-4)$ — the ruler pierces the top and bottom of the loop. Why this step? Two outputs for input $3$ is an instant failure.
3. Edge columns give one hit. At $x=5$: $y=\pm \sqrt{0}=0$, the single tangent point $(5,0)$. At $x=6$: $\sqrt{25-36}$ is imaginary — zero hits, the ruler misses the circle entirely. Why this step? Covering the degenerate columns shows I understand every vertical position, not just the easy middle.

Verify: $3^2+4^2=9+16=25$ ✓ and $3^2+(-4{)}^2=25$ ✓. Both points sit on the circle. ✗ Fails — not a function.

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Example 4 — Cells D & E: the two straightest lines

Forecast: One of these is the worst possible graph for the test; the other is a guaranteed pass. Which is which?

1. Analyse $x=4$ (Cell D). This graph is itself a vertical line. Place your ruler exactly on top of it: it overlaps at $(4,0),(4,1),(4,-7),\dots$ — infinitely many points. Why this step? The test asks "at most one hit per vertical line." Here one vertical line ($x=4$) hits infinitely often — the ultimate failure. The input $x=4$ maps to every real $y$ at once.
2. Analyse $y=2$ (Cell E). Fix $x=a$: the equation says $y=2$ regardless of $a$. Every column produces the single value $2$. Why this step? One $x$ → one $y=2$ every time; the definition of a function is satisfied with a flat rule.
3. Contrast them. $x=4$ is a relation but not a function of $x$; $y=2$ is a perfectly good (constant) function.

Verify: For $y=2$ at $x=100$, output is $2$ — one hit ✓. For $x=4$, the points $(4,0)$ and $(4,1)$ both satisfy $x=4$ — two (indeed infinite) hits. (a) Fails; (b) Passes.

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Example 5 — Cell F: piecewise with an endpoint clash

Forecast: The two rules overlap at exactly one $x$. Watch what each rule says there.

1. Evaluate both pieces at the shared point $x=2$. Top rule: $2+1=3$. Bottom rule: $5-2=3$. Why this step? Overlap points are where piecewise definitions can secretly assign two $y$-values. I must check whether both rules agree.
2. They agree. Both give $y=3$, so the vertical line $x=2$ hits only $(2,3)$ — a single point. Why this step? Agreement means one output, so the potential double-hit collapses into one.
3. Everywhere else only one rule applies, so at most one $y$ per $x$.

Verify: $2+1=3$ and $5-2=3$; the endpoints coincide. ✓ Passes.

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Example 6 — Cell G: a function with a gap

Forecast: There is a spot where this graph simply isn't there. Does a hole make it fail?

1. Where is it undefined? At $x=3$ the denominator is $0$; $h(3)$ does not exist. Why this step? The test only cares about columns where the graph exists. A missing column means zero hits, which is allowed.
2. Everywhere $x\ne 3$, one output. For any allowed $a$, $\frac{1}{a-3}$ is a single number. Why this step? One input → one output confirms the function property on the domain.
3. The vertical line $x=3$ hits nothing. Zero intersections ≤ one intersection, so the rule is not broken.

Verify: At $x=5$, $h=\frac{1}{2}=0.5$, one point. At $x=3$, undefined — no point to double up. ✓ Passes (see domain from graphs for reading the hole off the picture).

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Example 7 — Cell H: the scary wiggle

Forecast: It goes up, down, up, down forever. Surely something that busy fails? Guess first.

1. Fix $x=a$ and read off $y$. $\sin a$ is a single number for any $a$. Why this step? Oscillation is a left–right story (many $x$ share a $y$). The test is a up–down story (how many $y$ per one $x$). They are different directions.
2. Check a value. At $x=\frac{\pi }{2}\approx 1.571$, $\sin x=1$ — one hit. At $x=\pi \approx 3.1416$, $\sin x=0$ — one hit. Why this step? Sampling confirms the algebra: never two $y$ at one $x$.
3. Many $x$ can share the same $y$ (e.g. $\sin 0=\sin \pi =0$) — but that is fine; the vertical test never forbids repeated $y$-values.

Verify: $\sin (\pi /2)=1$ and $\sin (\pi )=0$, each a lone point in its column. ✓ Passes. (The wiggle does fail the horizontal line test — a different question about being one-to-one.)

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Example 8 — Cell I: word problem

Forecast: Real data — can a single instant have two temperatures?

1. Identify input and output. Input is time $t$; output is temperature $T$. The vertical line is "freeze the clock at one instant." Why this step? Naming which axis is the input tells me which direction the ruler slides.
2. Two readings at the same instant. $t=8$ gives both $T=19$ and $T=23$ — the vertical line $t=8$ hits the data twice. Why this step? Two outputs for one input is exactly the failing pattern.
3. Physical sanity. A single instant can only have one true temperature, so the double reading signals a logging error — if taken literally as data, it is not a function.

Verify (units): Input hours → output °C. At $t=8$ h we listed $19$ °C and $23$ °C, i.e. two distinct outputs for one input. ✗ Not a function (the two values $19\ne 23$ confirm the clash).

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Example 9 — Cell J: the exam twist (one equation, two questions)

Forecast: Same equation as Ex 2 — but the second question swaps the roles. Does the answer flip?

1. Part (a): $y$ as a function of $x$. Already done in Ex 2 — fixing $x=4$ gives $y=\pm 2$, two outputs. Fails. Why this step? Re-using the earlier result: the vertical-line direction tests $y$-per-$x$.
2. Part (b): swap the roles — treat $y$ as the input. Rearranged, $x=y^2$. Now fix $y=b$: $x=b^2$, a single number. Why this step? When $x$ is the output, the relevant test line is horizontal ($y=b$ fixed). Count how many $x$ per one $y$ — exactly one.
3. Conclusion. As written, $x=y^2$ is a function of $y$, even though $y$ is not a function of $x$.

Verify: For $y=3$, $x=3^2=9$ — one value. For $y=-3$, $x=9$ too (repeated output, still fine for a function). (a) Fails; (b) Passes.

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Recall One-line classifier

Given any graph, ask: is there a single upright ruler position that touches it 2+ times? Yes ::: Not a function. No (only ever 0 or 1) ::: It is a function.

Recall Fill the matrix from memory

Even power on $y$ (like $y^2=x$) ::: two branches — fails. A gap where the function is undefined ::: zero hits there — still passes. A vertical line $x=c$ as the whole graph ::: infinitely many hits — fails hardest.

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Connections

• Hinglish version of the parent
• 2.1.01-Definition-of-a-function — the "exactly one output" rule these examples enforce
• 2.2.01-Graph-of-a-function — reading graphs before testing them
• 2.2.05-Horizontal-line-test-for-injective-functions — the rotated question (Ex 7, Ex 9)
• 2.3.02-Domain-and-rangefrom-graphs — the gap in Ex 6
• 3.1.01-Inverse-functions — needs both tests to pass