What is a Function?

A function is a special relationship where each input (often called \( x \)) has exactly one output (\( y \)). Think of it as a machine: you put in a number, and the function gives you a result.

• Written as \( f(x) \), which means "the function of \( x \)"

Graphing Functions

We can show functions on a graph. The horizontal axis is usually \( x \), and the vertical axis is \( y \) or \( f(x) \).

Why Use Functions?

Functions describe patterns, rules, and relationships—like how distance changes over time, or how the price of candy depends on the number you buy.

Types of Functions

• Linear Functions: Straight lines (\( f(x) = mx + b \))
• Nonlinear Functions: Curved lines, like \( f(x) = x^2 \)

Reading Graphs

Graphs make it easy to see how changing the input changes the output.

Real-World Example

If a taxi costs $3 plus $2 per mile, the function is \( f(x) = 3 + 2x \), where \( x \) is the number of miles.