What a Function Does

How a function links an input to an output through a fixed rule, how to write that rule in symbols, and how to use it to answer questions.

--

Quantities in the world are rarely independent. Study longer and you cover more pages; heat water harder and it boils sooner. A function is the mathematical tool for capturing exactly this kind of link between two quantities.

Concept Input, rule, output
rule input output
A function takes something in, applies a fixed rule, and hands something back. The rule never changes; only what you feed it does.

One condition matters: each input gives exactly one output. Feed the same value in twice and you must get the same answer twice.

Concept A rule from everyday life

Suppose a student needs 5 minutes to study one page. That single sentence is a complete rule:

1 page takes 5 minutes.
4 pages take 20 minutes.
10 pages take 50 minutes.

The number of pages is the input, the time is the output, and "multiply by 5" is the rule connecting them.

Concept Writing it in symbols

Sentences are slow to work with, so the relationship is given letters:

Let p be the number of pages — the input.
Let t be the time in minutes — the output.
The function is t = 5p .

One short line now answers every question of this kind, for any number of pages you care to name.

Example Studying ten pages

How long does it take to study 10 pages?

The function is t = 5p .
Substitute p = 10 .
t = 5 \times 10 = 50 .
⟹ 50 minutes
Example A whole book

A textbook has 100 pages. How long would it take to study all of it?

Substitute p = 100 into t = 5p .
t = 5 \times 100 = 500 minutes.
500 \div 60 = 8 with 20 left over.
⟹ 500 minutes, or 8 hours and 20 minutes

The same one-line rule handled 10 pages and 100 pages without any change. That reusability is the whole point of writing it symbolically.

Example A function that decreases

Functions do not always grow. Consider boiling water: the hotter the flame, the sooner it boils.

A very high temperature — about 5 minutes.
A moderate temperature — about 7 minutes.
A low temperature — longer still.

Temperature is the input and boiling time the output, but here a larger input gives a smaller output. It is still a function: one input, one output.

Note Mistakes to avoid
Mixing up which quantity is the input and which is the output.
Expecting every function to increase — many decrease.
Allowing one input to produce two different outputs.
Substituting into the wrong variable in the formula.
Leaving the answer without its unit: 50 minutes, not just 50.
Summary
  1. A function is a rule linking an input to an output.
  2. Each input must give exactly one output.
  3. Naming the quantities turns a sentence into a formula such as t = 5p.
  4. Substituting a value into the formula answers the question directly.
  5. Outputs may rise or fall as the input grows; both are functions.
231. Arithmetic Sequences and Series
← Back to Semester