Functions as Rules That Map Inputs to Outputs

A function is a rule that assigns exactly one output to each input — no input can produce two different outputs at the same time. Think of the vending machine: you press button B3 and exactly one item drops. The machine never sends two different snacks for the same button press. That guarantee — one input, exactly one output, every time — is the whole idea. We write a function using notation like f(x), read "f of x," where x is the input. To evaluate f(3), you substitute 3 for x and compute the result. For example, if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7, and f(-1) = 2(-1) + 1 = -1. The notation tells you exactly which input you are plugging in. The same rule can be described three equivalent ways: • Table: two columns, one for input x and one for output f(x). Every x-value appears at most once. • Graph: a curve or scatter of points. The vertical line test tells you it is a function if every vertical line crosses the graph at most once — because a vertical line represents a single x-value, and if it hits two points the input has two outputs. • Formula: an algebraic expression such as f(x) = 2x + 1 or g(x) = x². Plug in x, get one number out. Here is the key misconception to avoid: a function CAN send two different inputs to the same output. f(x) = x² sends both x = 3 and x = -3 to 9. That is fine — multiple inputs can share one output. What is never allowed is one input mapping to two different outputs. A non-example: the equation x² + y² = 25 (a circle). For x = 3, y could be 4 or -4. That ambiguity means y is not a function of x here. Hint for the activity: when you examine a table, check that each input x appears only once. When you see function notation like g(-1), substitute -1 for x and simplify step by step.