Understanding Functions

What Is a Function?

A function is a rule that assigns to each input exactly one output.

$\text{Input} \xrightarrow{\text{function}} \text{Output}$

Function Notation

$f(x)$ means "the output of function $f$ when the input is $x$."

If $f(x) = 2x + 3$:

• $f(1) = 2(1) + 3 = 5$
• $f(4) = 2(4) + 3 = 11$
• $f(-2) = 2(-2) + 3 = -1$

Is It a Function?

Table test: Each input has only ONE output.

| Input | Output | Function? | |-------|--------|-----------| | 1 → 3, 2 → 5, 3 → 7 | ✅ Yes | Each input has one output | | 1 → 3, 2 → 5, 1 → 7 | ❌ No | Input 1 has two outputs |

Vertical Line Test: If any vertical line crosses the graph more than once, it's NOT a function.

Linear vs. Nonlinear Functions

Linear: Graph is a straight line, constant rate of change. $f(x) = mx + b$

Nonlinear: Graph is curved, rate of change varies. $f(x) = x^2, \quad f(x) = \sqrt{x}, \quad f(x) = |x|$

Comparing Functions

Functions can be represented as:

1. Equations: $y = 3x + 1$
2. Tables: pairs of $(x, y)$ values
3. Graphs: visual representation
4. Verbal descriptions: "3 times a number plus 1"

Compare functions by examining their rates of change (slopes) and initial values (y-intercepts).

Key idea: A function is like a machine — put in an input, get exactly one output. No input gives two different outputs!