1
Exactly one means at least one and at most one
The rule must produce an output for each input it claims to accept, and it must never produce two. Both halves can fail, and “Domain and Range” is largely about the first half.
Lesson · Lines, Functions and Graphs · Functions
What a Function Is
A function is a rule that assigns to each input exactly one output. The definition is short, the word doing the work is "exactly", and on a graph the whole condition can be checked by eye with the vertical line test.
each input → EXACTLY ONE output
two inputs may share an output (fine)
one input with two outputs (not a function)
vertical line test: no vertical line may cross twice
The definition, and what it does not forbid
Core ideas
What this lesson establishes
The first seven lessons used points and lines as solution sets. This lesson names the input-output idea that organises every graph that follows.
2
Many-to-one is allowed; one-to-many is not
f(x) = x² sends both 3 and −3 to 9, and that is a function. y² = x sends 4 to both 2 and −2, and that is not. The asymmetry is the definition, not a convention.
3
The vertical line test is the definition, drawn
A vertical line is the set of points sharing one x. If it meets a graph twice, that single input has two outputs. The test is not a separate fact to remember — it is what the definition looks like on paper.
The material
The definition and its test
What must hold, what is permitted, and how to check each representation.
Definition · Function
A function from a set of inputs to a set of outputs is a rule assigning to each input exactly one output. The input set is the domain; the outputs form the range.
Every word is load-bearing. "Each" forbids an input the rule cannot handle. "Exactly one" forbids an input with two answers. A rule that fails either is not a function, and calling it one anyway breaks everything downstream — notation in “Function Notation” assumes f(3) names a single number, and an inverse in “Inverse Functions” assumes the assignment can be run backwards.
What the definition permits
The asymmetry surprises people, so state it plainly: outputs may repeat, inputs may not. Several inputs sharing one output is ordinary and useful. One input with several outputs is a contradiction, because then "the" output of that input does not exist.
f(x) = x^2 x^2 + y^2 = 25
3 -> 9 x = 3 -> y = 4 AND y = -4
-3 -> 9 x = 0 -> y = 5 AND y = -5
two inputs, one output ONE input, two outputs
a function NOT a function
The circle is the clearest failure because it fails at almost every input, not at some awkward exception. It is a perfectly good curve and a perfectly good equation; it is simply not a function of x, which is why the course notes list conic sections as out of scope.
The vertical line test
Theorem · Vertical line test
A graph in the plane is the graph of a function of x if and only if no vertical line meets it more than once. A vertical line consists of all points with one fixed x, so two intersections are exactly one input with two outputs.
This is where “Graphing a Linear Equation”'s vertical line returns. The graph of x = 5 is itself a vertical line, so a vertical line meets it infinitely often: the input 5 has every number as an output. Of all the lines in this course, the vertical ones are precisely the non-functions — which is the same exception that blocked slope in “Slope” and slope-intercept form in “Slope-Intercept Form”, now explained rather than noted.
Checking the other representations
- A table. Look for a repeated input with different outputs. A repeated output is irrelevant.
- A set of pairs. Same test: two pairs with the same first coordinate and different second coordinates disqualify it.
- An equation. Solve for y if you can. If solving forces a ±, as with y² = x, the rule is not a function.
- A graph. The vertical line test.
Example · Six rules
y = 2x + 1 function. y = x² function. y = |x| function, since |−7| and |7| are both 7 and each input still gets one output. y² = x not a function. x = 5 not a function. x² + y² = 25 not a function.
One caution about the phrase "function of x". The circle fails as a function of x, and the vertical line x = 5 fails as a function of x while being a perfectly good function of y. The test is always relative to which variable is the input, and the vertical line test presumes the usual choice of x across and y up.
Interactive
Is it a function?
The vertical line test, drawn where it fails
Method
Deciding whether a rule is a function
Four questions, in this order. The first "no" settles it.
1
Identify the input
Usually x, but not always. Every later question depends on this choice, so make it explicit before testing anything.
2
Ask whether any input gets two outputs
In a table, look for a repeated input. In an equation, solve for the output and watch for a ± or an even root. On a graph, sweep a vertical line.
3
Ask whether any claimed input gets none
A rule that says nothing about some input in its stated domain is not a function on that domain. “Domain and Range” turns this into finding the domain.
4
If it fails, name the input that breaks it
"Not a function" is half an answer. "Not a function: x = 4 gives both 2 and −2" is the whole one, and it is the form that transfers to every later lesson.
Function or not
Score 0 / 0Choose the answer you can justify.
Six rules, classified with reasons
Give the verdict and the witness — the specific input that breaks the rule, when there is one.
1. y = 2x + 1 function one output per x
2. y = x^2 function -3 and 3 both give 9; allowed
3. y = |x| function -7 and 7 both give 7; allowed
4. y^2 = x NOT x = 4 gives y = 2 and y = -2
5. x = 5 NOT x = 5 gives every y at once
6. x^2 + y^2 = 25 NOT x = 3 gives y = 4 and y = -4
and a table:
x | 1 2 3 2 x = 2 appears twice,
y | 5 7 9 8 with 7 and 8 NOT a function
x | 1 2 3 4 every input once,
y | 5 5 5 5 outputs repeat function
Lines 2 and 3 are the ones to dwell on. Both send two different inputs to one output, and both are functions. The rule constrains what leaves an input, not what arrives at an output.
The two tables make the same point in the representation where it is easiest to misread. A repeated row of outputs looks suspicious and is fine; a repeated input with a different output is what disqualifies.
For a faded classification, use the pairs (−2, 1), (0, 3), (2, 1) and (0, 4). The supplied first move is to group them by their first coordinate. Finish the input-output table, give the verdict, and name the exact input and two outputs that witness it. Then remove one pair so the remaining relation becomes a function.
Common mistakes
1
Thinking repeated outputs break the definitiony = x² sends 3 and −3 to 9 and is a function. Only a repeated input with different outputs disqualifies a rule.
2
Applying the horizontal line test to decide functionhoodA horizontal line meeting a graph twice says the function is not one-to-one, which is a real property and the subject of “Inverse Functions”. It says nothing about whether the graph is a function.
3
Answering "not a function" without a witnessName the input that breaks it. Saying x = 4 gives both 2 and −2 is the answer that survives into “Domain and Range”'s domain work and “Inverse Functions”'s restrictions; the bare verdict does not.
Check your understanding
✓
Finish when you can apply the definition to a rule in any of its four representations and name the offending input whenever one exists.
Given equations, tables, sets of pairs and graphs mixed together, you should classify each and, for every failure, produce the input with two outputs. This is the definition the remaining seven lessons build on: notation, domain, transformation, composition and inverse all presuppose that an input has one answer.
Note: The definition explains an exception that has come up three times already. Vertical lines have no slope, cannot be written y = mx + b, and are not functions — all three because a single x is paired with many y. From “Function Notation” onward everything in this course is a function, and that is why the notation f(x) is about to be safe to use.