Lesson 41: Functions

New Concept

A function is a rule that connects inputs to outputs. For every single input, there is exactly one unique output—no exceptions!

What’s next

Next, you’ll explore how to represent functions using tables and graphs. We will see how to spot the difference between linear and proportional relationships.

Property

A function is a mathematical rule that identifies a relationship between two sets of numbers. For each input number there is one and only one output number.

Examples

• The perimeter of a square, $P = 4s$, is a function. An input of $s=3$ always gives the output $P=12$.
• The relationship shown in the table where $x=1$ maps to both $y=1$ and $y=-1$ is NOT a function because one input has two different outputs.
• The area of a circle, $A = \pi r^2$, is a function. A radius of $r=2$ will only ever produce an area of $A=4\pi$.

Explanation

Think of a function as a trusty machine. You put an input number in, the machine follows its one special rule, and it gives you a single, predictable output every time. No surprises, just consistent results!

Property

If all the input-output pairs of a function fall on a line, then the function is linear.

Examples

• The equation $y = x + 2$ is linear. Pairs like $(1, 3)$, $(2, 4)$, and $(3, 5)$ all line up perfectly.
• The function for the perimeter of an equilateral triangle, $P = 3s$, is linear because the points $(1,3)$ and $(2,6)$ are on the same line.
• The function for the area of a square, $A = s^2$, is not linear because the points $(1,1)$, $(2,4)$, and $(3,9)$ form a curve, not a line.

Explanation

Imagine you're earning money at a steady rate. For every hour you work, your pay goes up by the same amount. When you plot these points on a graph, they form a perfectly straight line—that's a linear function!

Property

A functional relationship is proportional if the ratio of the output to the input is constant. Its graph is a straight line that passes through the origin $(0, 0)$.

Examples

• The equation $y=2x$ is proportional. The ratio $\frac{y}{x}$ is always 2, and its graph goes through $(0,0)$.
• The relationship between pints and ounces is proportional ($o = 16p$). The ratio $\frac{o}{p}$ is always 16.
• The equation $y = x + 2$ is NOT proportional because its graph does not pass through the origin $(0,0)$.

Explanation

This is a special type of linear relationship! If you double the input, the output also doubles. The key is that the graph is a straight line that always starts from the very beginning, at point $(0,0)$.