A function is a mathematical relationship that connects inputs to outputs. Think of it like a machine: you put something in, and it gives you exactly one thing out. Each input value has only one corresponding output value.
Let's see some examples. For the function f of x equals 2x plus 1, when we input 3, we get output 7. For g of x equals x squared, input 4 gives output 16. The key rule is simple: one input always gives exactly one output.
To determine if something is a function, we use the vertical line test. Draw any vertical line on the graph. If it crosses the curve at most once, it's a function. If it crosses more than once, like with a circle, it's not a function because one input would have multiple outputs.
Function notation uses f of x to represent a function. For example, f of x equals 2x plus 1. To evaluate f of 3, we substitute 3 for x: f of 3 equals 2 times 3 plus 1, which equals 6 plus 1, giving us 7. So input 3 produces output 7.
To summarize what we've learned: A function is a special relationship where each input has exactly one output. We can test if something is a function using the vertical line test. Function notation helps us express these relationships clearly, and functions are essential tools throughout mathematics.