A function is a mathematical relationship that connects inputs to outputs. The key rule is that each input must have exactly one corresponding output. Think of it like a machine: you put something in, and you get exactly one thing out.
To determine if a relationship is a function, we use the vertical line test. If any vertical line intersects the graph more than once, it's not a function. This parabola is a function because every vertical line touches it at most once.
Functions use special notation. We write f of x equals 2x plus 1, where f is the function name, x is the input, and 2x plus 1 is the rule. For example, f of 3 equals 2 times 3 plus 1, which equals 7.
There are many types of functions. Linear functions create straight lines, quadratic functions form parabolas, and exponential functions show rapid growth or decay. Each type has unique properties that make them useful for modeling different real-world situations.
To summarize what we have learned: A function is a special relationship where each input has exactly one output. We can identify functions using the vertical line test. Function notation helps us express these relationships clearly. Different types of functions help us model various patterns in mathematics and real life.