A function is a special mathematical relationship between inputs and outputs. The key characteristic of a function is that each input value corresponds to exactly one output value. We can think of a function as a machine that takes an input from the domain and produces a unique output in the range.
To determine if a relationship is a function, we use the vertical line test. If any vertical line drawn through the graph intersects it at more than one point, then it is not a function. A true function will have each vertical line intersect the graph at most once, ensuring each input has exactly one output.
Function notation provides a clear way to express functions and evaluate them. We write f of x to represent the function, where f is the function name and x is the input variable. To evaluate a function, we substitute the input value for x. For example, if f of x equals 2x plus 1, then f of 3 equals 2 times 3 plus 1, which equals 7.
There are many different types of functions, each with unique characteristics. Linear functions create straight lines and have the form f of x equals mx plus b. Quadratic functions form parabolas with the form f of x equals ax squared plus bx plus c. Exponential functions show rapid growth or decay with the form f of x equals a to the power of x. Absolute value functions create V-shaped graphs.
To summarize what we have learned about functions: A function is a special relationship where each input corresponds to exactly one output. We can verify if a relationship is a function using the vertical line test. Function notation provides a clear way to express and evaluate functions. There are many types of functions, each creating unique patterns when graphed. Functions are fundamental mathematical tools used throughout mathematics and science.