Functions
Functions: Domain, Range and Notation
The one-output rule that defines a function, the vertical line test, function notation, and what domain and range actually name.
What this page covers
- State the defining property of a function
- Apply the vertical line test to a graph
- Evaluate a function at a value, including a negative or an expression
- Distinguish domain from range and dependent from independent variable
What a function is
A rule that assigns to each element of one set exactly one element of another. The source states the special property directly: for each value in the domain there is exactly one value in the range.
The source pictures it as a number-cruncher: values go in, the rule acts, values come out. The set of admissible inputs is the domain; the set of outputs produced is the range.
The source's arrow-diagram example
With the rule x → x2 and inputs {1, 2, 3}:
| x | 1 | 2 | 3 |
|---|---|---|---|
| y = x2 | 1 | 4 | 9 |
The source distinguishes the range from the larger set the outputs are drawn from — here E = {0, 1, 2, …, 9}. The range is what is actually hit; the codomain is what is available.
A relation is any set of ordered pairs. The typed notes make the distinction with a small example: {(1, 4), (3, 5), (5, 5), (7, 11)} is a function, because no two pairs share a first coordinate. Adding (3, 9) would break it, since 3 would then have two outputs.
(3, 5) and (5, 5) coexist happily. The rule constrains inputs, not outputs.
The vertical line test
A graph represents a function if and only if no vertical line meets it more than once.
It is the definition translated into a picture. A vertical line is the set of points with a given x; meeting the graph twice means that x has two outputs.
| Relation | Function? | Reason |
|---|---|---|
| y = x2 | Yes | Each x gives one y |
| x = y2 | No | At x = 4, y = 2 and y = -2 |
| x2 + y2 = 9 | No | The source draws a vertical line cutting the circle twice |
| 2x - 3y = 5 | Yes | Solves to y = 2x - 53, one value per x |
The source works the last two together, contrasting 2x - 3y = 5, which describes a function, with x2 + y2 = 9, which does not. Both are perfectly good equations; only one is a function.
Function notation
Writing y = f(x) names the rule so it can be referred to. The source lists the common letters — f, g, h, F — and warns that y = f(x) and y = F(x) are different functions, not the same one.
The source underlines this: f(x) means the value of f at x, not f times x. So f(3 - 2) means f(1), and not 3f - 2f. The notation is unfortunate but universal, and the habit of reading it correctly has to be built deliberately.
An alternative notation the source gives is the arrow form, f : x → √5 - x, which reads as f sends x to √5 - x and avoids the ambiguity entirely.
Evaluating a function
Substitute the given value for the variable, everywhere it occurs, and simplify.
Worked examples from the source
For f(x) = √6 - x:
6 - (-10) = 16, not -4. When the input is negative, substituting without brackets produces a sign error almost every time.
Substituting an expression
For f(x) = x2 + 2:
The input need not be a number. Substituting -x is how a function is tested for evenness, which the source goes on to do.
Dependent and independent variables
When y = f(x), the value of y depends on the choice of x. So x is the independent variable and y the dependent one.
The source's example — area of a circle
Here r is independent and A dependent. But the source makes a further point: if A is known and r is wanted, the roles reverse.
Which variable is independent is a matter of what is known and what is sought, not a property of the formula. The condition r > 0 comes from context: a radius cannot be negative, so the negative root is discarded.
The source also gives a data example: maximum temperature plotted against day. Each day has exactly one maximum, so the relation is a function even though no formula describes it. A function need not be given by an equation.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Reading f(x) as f times x | It is the value of f at x | f(3 - 2) = f(1) |
| f(-10) = √6 - 10 | √6 + 10 = 4 | Bracket the substituted value |
| Calling x = y2 a function | It is a relation | Vertical line test fails |
| Thinking two inputs may not share an output | They may | x2 sends 2 and -2 to 4 |
| Confusing range with codomain | The range is what is actually reached | x2 on ℝ has range [0, ∞) |
| (x + h)2 = x2 + h2 | x2 + 2xh + h2 | The middle term is not optional |
Frequently asked questions
What is the difference between a relation and a function?
A relation is any set of points. A function is a relation in which each input has exactly one output. Every function is a relation; most relations are not functions.
Does f(x) mean f times x?
No, and the source flags this directly. f(x) is the value of the function f at input x. The notation is unfortunate but universal.
Can two inputs share an output?
Yes. f(x) = x2 sends both 2 and -2 to 4, and it is still a function. The rule forbids one input having two outputs, not two inputs sharing one.
What is the range?
The set of values the function actually produces. It is often harder to find than the domain, because it requires knowing what the function can reach, not just what it accepts.
Source. Handwritten teaching notes, Week 5, pages 1-3, with the typed supplementary notes on the domain of functions.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.