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Identifying Functions

Decide whether a set of ordered pairs, a table, a graph, an equation, or a real-life rule is a function, using the one-output rule and the vertical line test.

Practice this skill 25 min Identify functions Leer en español
Key idea

What you'll learn

A function is a rule that gives exactly one output for each input. In this lesson you will learn to spot functions — and non-functions — in every form the GED uses:

  • a list of ordered pairs,
  • a table of values,
  • a graph (using the vertical line test),
  • an equation,
  • a real-life description ("each employee → his or her ID number").

Why it matters on the GED. You will see questions such as "Which table represents a function?", "Click the graph that is not a function," or "For which value of k is this relation not a function?". These are quick points if you know the rule — and easy points to lose if you check the wrong column.

Why it matters in real life. A vending machine is a function: press B4 and you always get the same snack. If pressing B4 sometimes gave chips and sometimes gave candy, it would not be a function — and you would be annoyed! Functions describe reliable, predictable rules: price → tax, hours → pay, temperature in °F → temperature in °C.

Words to know

  • Relation — any set of input–output pairs. Example: \{(1, 4), (2, 5), (2, 7)\}.
  • Function — a relation in which every input has exactly one output. Example: \{(1, 4), (2, 5), (3, 5)\}.
  • Input — the first number of an ordered pair, the x-value. In (3, 8) the input is 3.
  • Output — the second number, the y-value. In (3, 8) the output is 8.
  • Ordered pair — (x, y); order matters: (2, 5) \ne (5, 2).
  • Domain — the set of all inputs. For \{(1,4), (2,5), (3,5)\} the domain is \{1, 2, 3\}.
  • Range — the set of all outputs. For the same relation the range is \{4, 5\}.
  • Vertical line test — if any vertical line crosses a graph more than once, the graph is not a function.
  • Mapping — arrows from inputs to outputs, like "Ana → 85."

All Math words in the glossary

1. The one rule: each input gets exactly one output

Look at the inputs (x-values). The question is always: does any input appear with two different outputs?

  • If no input repeats with a different output → function.
  • If some input is paired with two (or more) different outputs → not a function.

Repeated outputs are fine. Many inputs can share one output. Every bag of the same chips costs $1.50 — different bags, same price. That is still a function.

RelationRepeated input with different outputs?Function?
\{(1, 4), (2, 5), (3, 6)\}noneYes
\{(1, 3), (2, 3), (3, 3)\}none (only the output 3 repeats)Yes
\{(2, 5), (4, 7), (4, 9)\}input 4 → 7 and 9No
\{(5, 1), (5, 1), (6, 2)\}input 5 → 1 both times (same pair listed twice)Yes

The same relations as mapping diagrams:

Three mapping diagrams with inputs on the left and outputs on the right. First: 1 to 4, 2 to 5, 3 to 6 — a function. Second: 1, 2 and 3 all point to 3 — still a function, because only the output repeats. Third: 2 to 5, and the input 4 points to both 7 and 9 (shown in red) — not a function. Caption: many inputs may share one output; one input may not have two outputs.InputOutput123456{(1, 4), (2, 5), (3, 6)}FunctionInputOutput1233{(1, 3), (2, 3), (3, 3)}FunctionInputOutput24579{(2, 5), (4, 7), (4, 9)}Not a functionMany inputs may share one output. One input may NOT have two outputs.

Tables work the same way. In a table with an x row and a y row, scan the x row for repeats. If an x-value appears twice, look at the y-values under it. Same y → still a function; different y → not a function.

Real-life descriptions. Ask: "If I know the input, do I know the output for sure?"

  • Input: a person; output: their birthday → function (each person has one birthday).
  • Input: a birthday; output: a person → not a function (many people share a birthday).
  • Input: gallons of gas at $3.60/gal; output: cost → function.
  • Input: age in years; output: height → not a function (two 30-year-olds can have different heights).

2. The vertical line test for graphs

A point (x, y) on a graph sits directly above or below its input x. If two points share the same x, they lie on the same vertical line. So:

Vertical line test: imagine sliding a vertical line (like a ruler held straight up) across the graph. If it ever touches the graph at two or more points at once, the graph is not a function. If it never touches more than once, the graph is a function.

Vertical line test. Graph A: an upward parabola; a dashed vertical line at x = 2 crosses it at exactly one point, so Graph A is a function. Graph B: a circle of radius 3 centered at the origin; a dashed vertical line at x = 1 crosses it at two points, about (1, 2.8) and (1, -2.8), so Graph B is not a function.−4−3−2−11234−4−3−2−11234xyGraph A−4−3−2−11234−4−3−2−11234xyGraph B

In Graph A (a parabola), every vertical line hits the curve at most once → function. In Graph B (a circle), the line x = 1 hits the circle at about (1, 2.8) and (1, -2.8) → not a function.

Shapes you should know on sight:

GraphFunction?Why
Non-vertical straight lineYeseach x has one point
Horizontal line y = 2Yesevery input has the output 2
Vertical line x = 2Noinput 2 has infinitely many outputs
Parabola opening up or downYes
Parabola opening sidewaysNoa vertical line cuts it twice
Circle or ovalNo
V-shape (absolute value), S-shaped cubicYes

Scatter plots (separate dots): look for two dots stacked on the same vertical line. Points (2, 3) and (2, -1) share x = 2 → not a function.

Do not use a horizontal line test. A horizontal line hitting a graph twice just means two inputs share an output — that is allowed.

3. Functions from equations

When a relation is given as an equation, ask: if I choose an x, is there exactly one y?

  • y = 3x - 2: pick x = 4 → y = 10 only. Function. Every equation of the form y = mx + b is a function.
  • y = x^2: x = 3 → y = 9 only. Function. (Note: x = -3 also gives 9. Two inputs, same output — allowed.)
  • y = 7: every input gives 7. Function (horizontal line).
  • x = 4: the input 4 goes with every y. Not a function (vertical line).
  • x = y^2: for x = 9, y can be 3 or -3. Not a function (sideways parabola).
  • x^2 + y^2 = 25: for x = 3, y^2 = 16, so y = 4 or -4. Not a function (circle).
Four small graphs with dashed red vertical test lines. Top left: y = 7, a horizontal line; every vertical line meets it once, so it is a function. Top right: x = 4, a vertical line; the input 4 is paired with many outputs such as (4, 2) and (4, 4), so it is not a function. Bottom left: x = y squared, a parabola opening sideways; the vertical line x = 9 meets it at (9, 3) and (9, −3), so it is not a function. Bottom right: the circle x squared plus y squared = 25; the vertical line x = 3 meets it at (3, 4) and (3, −4), so it is not a function.−4−224−224680xyy = 7 (horizontal)Function: each vertical line hits once−2246−4−2240xy(4, 2), (4, 4), …x = 4 (vertical)Not a function: input 4 has many outputs−2246810−4−2240xy(9, 3)(9, −3)x = y²Not a function: x = 9 gives y = 3 or −3−6−4−2246−6−4−22460xy(3, 4)(3, −4)x² + y² = 25Not a function: x = 3 gives y = 4 or −4

Shortcut: try to solve for y. If you are forced to write "\pm" (plus-or-minus), or if y is missing completely (like x = 4), it is not a function. If you get one clean expression in x, it is a function.

Function notation. When a relation is a function, we often name it f and write f(x) ("f of x") for its output. f(x) = 3x - 2 means the same thing as y = 3x - 2. You will practice evaluating f(x) in the next lesson.

Formula

Rules to memorize (not formulas)

There is no formula on the GED formula sheet for this skill. Memorize these two rules instead:

\text{Function} \iff \text{each input } x \text{ has exactly one output } y \text{Graph is a function} \iff \text{no vertical line meets it more than once}
  • Input = x = first number in (x, y) = top row (or left column) of a table.
  • Output = y = second number = bottom row (or right column).

Not on the formula sheet — these are ideas you must know by heart.

Worked example

Example 1 (easy): ordered pairs

Is the relation \{(3, 1), (5, 2), (-1, 4), (5, -2), (0, 1)\} a function?

  1. List the inputs: 3, 5, -1, 5, 0.
  2. The input 5 appears twice.
  3. Look at its outputs: (5, 2) and (5, -2). The outputs are different (2 and -2).
  4. The output 1 appears twice (with inputs 3 and 0) — that is fine.
Left: a mapping diagram of the relation {(3, 1), (5, 2), (−1, 4), (5, −2), (0, 1)}. Inputs −1, 0, 3, 5; outputs −2, 1, 2, 4. Arrows: 3 to 1, 0 to 1, −1 to 4, and the input 5 (in red) has two arrows, to 2 and to −2. Right: the same five points plotted; the dashed vertical line x = 5 passes through both (5, 2) and (5, −2), so the relation is not a function.InputOutput−1035−2124Input 5 has two arrows−2−1123456−3−2−1123450xy(3, 1)(−1, 4)(0, 1)(5, 2)(5, −2)x = 5 meets two points → not a function

Answer: Not a function, because the input 5 is paired with both 2 and -2.

Check: If I ask "what is the output when x = 5?", there is no single answer — so it fails the definition ✓

Worked example

Example 2: tables

Which table represents y as a function of x?

Table 1

x1234
y7777

Table 2

x0112
y2345
  1. Table 1: the inputs 1, 2, 3, 4 are all different. Each has one output (always 7). Repeated outputs are allowed → function.
  2. Table 2: the input 1 appears twice, with outputs 3 and 4 → not a function.
Two tables, each above its graph. Table 1: x = 1, 2, 3, 4 and y = 7, 7, 7, 7; its four points lie in a horizontal row and each vertical line meets only one point, so it is a function. Table 2: x = 0, 1, 1, 2 and y = 2, 3, 4, 5, with the repeated input 1 highlighted in red; the points (1, 3) and (1, 4) lie on the same dashed vertical line x = 1, so it is not a function.Table 1x1234y7777Table 2x0112y234501234502468xyevery output is 7Function ✓ (each x once)01234502468xy(0, 2)(2, 5)(1, 3)(1, 4)Not a function ✗ (x = 1 twice)

Answer: Table 1 is a function; Table 2 is not.

Check: Plot Table 2's points (1, 3) and (1, 4): they lie on the same vertical line x = 1, so the vertical line test fails ✓

Worked example

Example 3: graphs and the vertical line test

Which of these graphs are functions? A: an upward parabola; B: a circle; C: a V-shape y = \lvert x \rvert - 1; D: the horizontal line y = 2.

Four graphs on coordinate planes from -4 to 4. Graph A: a parabola opening upward with vertex at (0, -3). Graph B: a circle of radius 3 centered at the origin. Graph C: a V-shaped graph y = |x| - 1 with vertex at (0, -1). Graph D: a horizontal line y = 2.−4−3−2−11234−4−3−2−11234xyGraph A−4−3−2−11234−4−3−2−11234xyGraph B−4−3−2−11234−4−3−2−11234xyGraph C−4−3−2−11234−4−3−2−11234xyGraph D
  1. A (parabola): any vertical line crosses it once → function.
  2. B (circle): the vertical line x = 0 crosses at (0, 3) and (0, -3) → not a function.
  3. C (V-shape): each vertical line crosses once → function.
  4. D (horizontal line): each vertical line crosses once → function.

Answer: A, C and D are functions; B is not.

Check: Find one input with two outputs on B: x = 0 gives y = 3 and y = -3 ✓ No such input exists on A, C or D.

Worked example

Example 4: real-life relations (word problem)

A teacher's grade book shows: Ana → 85, Ben → 92, Cara → 85, Dev → 78. Is student → score a function? Is score → student a function?

  1. Student → score: each student has exactly one score. Cara and Ana both have 85, but that is a repeated output → function.
  2. Score → student: now the score is the input. The input 85 goes to two students (Ana and Cara) → not a function.
Two mapping diagrams of the grade book. Left, student to score: Ana to 85, Ben to 92, Cara to 85, Dev to 78. Each student has one arrow, so it is a function even though two arrows end at 85. Right, score to student: 85 (in red) points to both Ana and Cara, 92 to Ben, 78 to Dev. The input 85 has two outputs, so it is not a function.StudentScoreAnaBenCaraDev788592student → scoreFunction ✓ScoreStudent788592AnaBenCaraDevscore → studentNot a function ✗ (85 → 2 names)

Answer: student → score is a function; score → student is not.

Check: Ask "Who scored 85?" — there are two answers, so the reversed relation fails ✓

Worked example

Example 5 (GED-hard): work backwards with a missing value

The relation \{(-2, 5), (0, 3), (k, 1), (4, -1)\} is not a function. Which values could k be?

  1. The relation fails only if an input repeats with a different output.
  2. The pair (k, 1) has output 1. The other outputs are 5, 3 and -1 — all different from 1.
  3. So if k equals any existing input (-2, 0, or 4), that input would have two different outputs.
    • k = -2: (-2, 5) and (-2, 1) ✗
    • k = 0: (0, 3) and (0, 1) ✗
    • k = 4: (4, -1) and (4, 1) ✗
Coordinate plane with the points (−2, 5), (0, 3) and (4, −1). A dotted horizontal line shows the output 1. Hollow red circles at (−2, 1), (0, 1) and (4, 1), each joined by a dashed vertical line to the existing point with the same x, show that k = −2, 0 or 4 would give one input two outputs. A hollow green circle at (1, 1) shows that k = 1 uses a new input and keeps the relation a function.−4−3−2−1123456−22460xy(−2, 5)(0, 3)(4, −1)k = −2 ✗k = 0 ✗k = 4 ✗k = 1 ✓y = 1(k, 1) breaks thefunction if it sharesan x with another pointRed: same x → ✗Green: new x → ✓

Answer: k = -2, 0, or 4. Any other value keeps it a function.

Check: Try k = 1: inputs -2, 0, 1, 4 are all different → function, as expected ✓

Calculator tip

This skill is about reasoning, not arithmetic, so the calculator rarely helps. The TI-30XS MultiView cannot draw graphs.

  • Use it only when an equation needs a quick value — for example, to check whether x^2 + y^2 = 25 gives two y-values at x = 3: compute 25 - 3^2 (use the x² key) to get 16, then 2nd + x² (√) to get 4. Both 4 and -4 work, so it is not a function.
  • For tables, ordered pairs, and graphs, your eyes and the vertical line test are faster than any keystroke. Treat these as calculator-free questions.
Common mistake

Common mistakes

  • Checking the outputs instead of the inputs. ✗ "The y-values 3 and 3 repeat, so it is not a function." ✓ Repeated y-values are fine; only a repeated x with different y-values breaks a function.
  • Using a horizontal line test. ✗ "A horizontal line crosses the parabola twice, so it is not a function." ✓ Only vertical lines matter. Parabolas opening up or down are functions.
  • Calling horizontal lines non-functions. ✗ "y = 5 has no x, so it is not a function." ✓ y = 5 gives one output (5) for every input — it is a function. It is the vertical line x = 5 that fails.
  • Thinking an identical repeated pair breaks the rule. ✗ (3, 8) listed twice → "not a function." ✓ Input 3 still has only one output, 8.
  • Thinking a function needs a pattern or even spacing. ✗ "The x-values jump 1, 4, 10, so it is not a function." ✓ The inputs can be any numbers; the outputs need no pattern. Only the one-output rule matters.
  • Mixing up (x, y) order. In (5, 2) the input is 5, not 2.
Test strategy

GED test strategy

  • Scan the inputs only. For ordered pairs or tables, run your finger along the x-values first. No repeats → it is a function — you are done in seconds.
  • For graphs, physically imagine the ruler. Look for any place where the curve doubles back over itself (a circle, a sideways "C," a vertical line).
  • "NOT a function" questions: read the question twice — on the GED it is easy to answer the opposite of what was asked. Underline the word not on your erasable note board.
  • Equations: if you see y^2 (without a y to the first power to solve cleanly) or x = \text{number}, suspect "not a function."
  • Eliminate answer choices that give the right verdict for the wrong reason (for example, "Yes, because the y-values increase").
  • Time: these are typically 30–60 second questions; bank the time for harder problems.

Check yourself

Identify functionsNo calculator

Which set of ordered pairs represents a function?

Answer choices
Hint

Look only at the first number (the input) of each pair. Does any input show up with two different outputs?

Check yourself

Identify functionsNo calculatorHot spot

The four graphs below show relations between x and y.

Click the graph that does NOT represent y as a function of x.

Click on the figure to mark your answer (or focus it and use the arrow keys).

Four graphs on coordinate planes from -4 to 4. Graph A: a parabola opening upward with vertex at (0, -3). Graph B: a circle of radius 3 centered at the origin. Graph C: a V-shaped graph y = |x| - 1 with vertex at (0, -1). Graph D: a horizontal line y = 2.−4−3−2−11234−4−3−2−11234xyGraph A−4−3−2−11234−4−3−2−11234xyGraph B−4−3−2−11234−4−3−2−11234xyGraph C−4−3−2−11234−4−3−2−11234xyGraph D
Hint

Imagine sliding a vertical ruler across each graph. On which graph does the ruler touch two points at once?

Summary

  • A function gives exactly one output for each input; repeated outputs are allowed.
  • Ordered pairs / tables: look for a repeated x with different y-values — that breaks the function.
  • Graphs: use the vertical line test; circles, sideways parabolas and vertical lines fail; horizontal lines pass.
  • Equations: if solving for y gives \pm or no y at all (x = 4), it is not a function.
  • Real life: ask "If I know the input, do I know the output for sure?"
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