Exploring Functions and Their Graphs

A function is a special relationship where each input (often called \( x \)) has exactly one output (often called \( y \)). You can think of a function as a machine: put in a number, get out another number based on a rule.

The Rule of the Machine

If the rule is \( y = 2x + 1 \), then for each value of \( x \), you get a value of \( y \):

• If \( x = 1 \), then \( y = 2(1) + 1 = 3 \).
• If \( x = 3 \), then \( y = 2(3) + 1 = 7 \).

Graphing Functions

When you plot the input and output pairs on a coordinate grid, you get a graph. For linear functions like \( y = mx + b \), the graph will be a straight line.

Why Functions Matter

Functions help us describe patterns, make predictions, and understand relationships in science, business, and everyday life.

Quick Tips

• The domain is the set of possible inputs.
• The range is the set of possible outputs.