A function is a special relationship between inputs and outputs. The key rule is that each input has exactly one corresponding output. Think of it like a machine: you put something in as an input, and you get exactly one thing out as an output. For example, if we have the function f of x equals x squared, and we input 3, we get exactly one output: 9.
To determine if a relationship is a function, we use the vertical line test. A function means each input has exactly one output. Look at the green curve: for any x value, there's only one y value. But the red curve fails the test because one x value gives two y values. When we draw a vertical line, it should intersect the graph at most once for it to be a function.
Function notation uses f of x to represent a function. Here, f is the function name, x is the input variable, and f of x is the output value. For example, if f of x equals 2x plus 1, then f of 3 equals 2 times 3 plus 1, which equals 7. Let's see how different inputs produce different outputs using this notation.
There are many types of functions. Linear functions have the form f of x equals mx plus b and create straight lines. Quadratic functions have the form f of x equals ax squared plus bx plus c and create parabolas. Exponential functions have the form f of x equals a to the power of x and show rapid growth or decay. Each type has its own unique shape and properties.
To summarize what we have learned about functions: A function is a special relationship that maps each input to exactly one output. We can use the vertical line test to determine if a graph represents a function. Function notation like f of x helps us express input-output relationships clearly. Different types of functions create different shapes on graphs, and functions are fundamental tools used throughout mathematics.