Functions and Graphs

## 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.