What is Linear function? — Mathematics

Definition 67.1 Primary & Middle School Mathematics · Chapter 67 — Linear and Affine Functions

A linear function multiplies every number by a fixed number $a$ :

f(x) = a x .

The number $a$ is the coefficient of $f$ . Two quantities are proportional exactly when one is a linear function of the other.

Two linear functions: both graphs pass through the origin; the coefficient a is the height reached at x = 1. — Two linear functions: both graphs pass through the origin; the coefficient $a$ is the height reached at $x = 1$ .

Examples

Example 67.2

If apples cost $3$ per kilogram, the price of $x$ kilograms is $f(x) = 3x$ : buying twice as much costs twice as much, and the price of $2.5$ kg is $f(2.5) = 7.5$ .

Conversely, if a linear function satisfies $f(4) = 10$ , then $a = \frac{10}{4} = 2.5$ and $f(x) = 2.5x$ : one value determines a linear function completely.

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