Functions and the Coordinate Plane
A function is a rule that assigns exactly one output to each input. Functions are the central concept of all higher math. Every function can be represented as an equation, a table, a graph, or a mapping diagram.
Key Concepts
What Is a Function?
A function maps each input (x) to exactly one output (y). The equation y = 2x + 1 is a function: every x gives exactly one y. A relation where x = 4 maps to both y = 3 and y = 7 is NOT a function. The vertical line test: if any vertical line crosses the graph more than once, it is not a function.
Function Notation
f(x) is read 'f of x.' It means the output when the input is x. f(x) = 3x − 2. f(4) = 3(4) − 2 = 10. f(0) = −2. The domain is all allowed inputs; the range is all possible outputs.
Tables and Graphs
Make a table: pick x values, calculate y = f(x), plot (x, y) points, connect them. A linear function makes a straight line. The rate of change (how fast y changes compared to x) is the slope.
🆕 Function Grapher
Enter a linear function (y = mx + b) and see the table and graph.
✅ Check Your Understanding
1. What makes a relation a function?
2. f(x) = 4x − 3. What is f(2)?
3. What is the vertical line test for?