SECONDARY MATHEMATICS · FUNCTIONS BEFORE FORMAL FUNCTIONS
A graph can represent a perfectly valid mathematical relation without representing y as a function of x. The deciding question is whether each permitted x chooses exactly one y.
One input, one output
The graph y = x² represents y as a function of x. For any real x, there is exactly one corresponding y. x = 2 gives y = 4; x = −2 also gives y = 4. Sharing an output is allowed.
The problem occurs when one x-value corresponds to two or more y-values in the same relation.
The vertical-line test
Imagine moving a vertical line across the graph. If any vertical line intersects the graph more than once, then at that x-value there is more than one y-value. The graph does not represent y as a function of x over that relation.
If every vertical line meets the graph at most once, the graph passes the test.
Why a circle fails
x² + y² = 25 describes a circle. At x = 0, y can be 5 or −5. Therefore the full circle does not define y as a function of x.
This does not make the circle invalid mathematics. It means the relation has two y-values for some x-values.
Half a circle can be a function
The upper semicircle y = √(25 − x²), with −5 ≤ x ≤ 5, gives exactly one nonnegative y for each permitted x. It is a function of x. The lower semicircle y = −√(25 − x²) is also a function on the same x-domain.
Restricting a relation can change whether it defines a function.
A sideways parabola
x = y² gives y = ±√x for x ≥ 0. For x = 4, y can be 2 or −2, so the full sideways parabola is not y as a function of x.
However, x can be a function of y: each real y produces exactly one x = y². Function status depends on which variable is treated as the input.
Vertical lines are functions
The equation x = 3 is a vertical line. As a relation in the coordinate plane it is valid, but it does not express y as a function of x because x = 3 corresponds to infinitely many y-values.
By contrast, y = 3 is a horizontal line and is a function of x: every permitted x maps to the single output 3.
Do not confuse “function” with “one-to-one”
y = x² is a function even though x = 2 and x = −2 both produce y = 4. A function may send different inputs to the same output. A one-to-one function imposes the stronger condition that different inputs have different outputs.
Tables can fail the same test
If a table contains input 2 paired with output 5 and also input 2 paired with output 7, it does not describe a function unless those rows refer to different rules or conditions that have not been represented. If inputs 2 and 3 both map to 5, the table can still represent a function.
Context can make the input choice important
A route may have distance as a function of elapsed time under a particular motion model, while time may not be uniquely determined by distance if the traveller passes the same location more than once. The variables’ physical meanings matter.
Worked classifications
A. y = 2x + 1: function of x.
B. y = |x|: function of x.
C. x² + y² = 9: not y as a function of x over the full circle.
D. x = 4: not y as a function of x.
E. y = 4: function of x.
A careful graph-reading routine
- State which variable is the input.
- Identify the domain being considered.
- Check whether any permitted input has more than one output.
- Use the vertical-line test when x is the input.
- If it fails, ask whether a sensible restriction produces a function.
Continue
Build from Input, Output and Dependency, Four Representations and Domain and Range. Return to the Secondary Mathematics Master Index.