Varying and Constant Functions

Independent and dependent variables, the one-output rule, and how to tell varying functions, constant functions and non-functions apart using the vertical line test.

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Not every relationship between two quantities deserves the name function. There is one condition that must hold, and once you know it you can sort any relationship into three groups: varying, constant, or not a function at all.

Concept Independent and dependent variables

Two quantities in a relationship do not have equal standing.

The independent variable is the one you choose or control.
The dependent variable is the one that responds to that choice.

When water is heated, time is independent — it advances whatever you do — and temperature is dependent, because it follows from how long the heating has gone on.

Concept The rule that decides everything

One input must give one output — never two.

A single question may only have a single answer. If one input value could produce two different results at once, the relationship is not a function.

Type 1 Varying function

Each input gives its own distinct output, so the graph climbs or falls as you move along it.

Heating water: every moment in time has its own temperature.
The graph is a rising or falling curve.
⟹ this is a function
Type 2 Constant function

The output stays the same no matter what the input is, giving a flat horizontal graph.

A still tank: the water level reads the same at every moment.
The graph is a horizontal line.
⟹ this is still a function

It is easy to assume a function must change. It need not. The requirement is one output per input — and an unchanging output satisfies that perfectly.

Type 3 Not a function

If one input yields two outputs simultaneously, the relationship fails the test.

A single instant showing two different readings at once.
The graph contains a vertical line or a full circle.
⟹ this is not a function
Concept Reading it off the graph
varying constant not a function
Sweep a vertical line across the graph. If it ever meets the curve at two points, one input is producing two outputs — so the graph does not represent a function.
Example Classifying three situations
Water being heated — temperature rises steadily, a varying function.
A still tank — the level never moves, a constant function.
One instant with two readings — not a function.
⟹ the first two qualify; the third does not

Only the third breaks the rule, and it breaks it for the one reason that matters: a single input with more than one output.

Note Mistakes to avoid
Ruling out a constant function because the output never changes.
Swapping the independent and dependent variables.
Accepting a graph in which a vertical line meets the curve twice.
Thinking two different inputs may not share an output — they may.
Judging by the shape of the graph rather than by the one-output rule.
Summary
  1. The independent variable is chosen; the dependent one responds.
  2. A function must give exactly one output for each input.
  3. A varying function gives each input its own distinct output.
  4. A constant function keeps one output throughout and is still a function.
  5. If a vertical line meets the graph twice, it is not a function.
231. Arithmetic Sequences and Series
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