Functions: FAQ

A function is a mathematical rule that matches inputs to outputs. We can think of it like a machine: put a number in, the machine does some calculations, and out pops a corresponding number.

Inputs and outputs don't have to be numbers. Functions themselves can be inputs and outputs.

Learn more with our What is a function video.

Functions are used in all sorts of real-world applications! For example, we use functions to model physical processes, like the motion of a car or the growth of a population. We can also use them to analyze data, like finding the rate of change of a company's profits over time.

Practice with our Graph interpretation word problems exercise.

The domain is the set of all possible inputs that the function can take, while the range is the set of all outputs the function can produce.

Practice with our Domain and range from graph exercise.

To be a function, every input must correspond to exactly one output. If we find an input that has two different outputs, then we know that the relationship is not a function.

Practice with our Recognize functions from graphs exercise.

Practice with our Recognize functions from tables exercise.

Function notation is a shorthand way of expressing a function. Instead of writing out the equation every time, we can use a letter (usually $f$f) to represent the function, and write $f(x)$f, left parenthesis, x, right parenthesis to indicate that we are plugging in the value $x$x as the input.

Practice with our Function notation word problems exercise.

An absolute maximum or minimum is the highest or lowest point on the entire function. A relative maximum or minimum is the highest or lowest point within a given interval, but not necessarily the entire function. These points can be important for understanding the behavior of a function.

Practice with our Relative maxima and minima exercise.

Practice with our Absolute maxima and minima exercise.

A function is increasing over an interval if the outputs get larger as the inputs increase. A function is decreasing if the outputs get smaller as the inputs increase.

Practice with our Increasing and decreasing intervals exercise.

The average rate of change is a measure of how much a function changes over a given interval. To calculate it, we divide the change in output by the change in input.

Practice with our Average rate of change: graphs & tables exercise.

Inverse functions are functions that "undo" each other. If we plug the output of one function into the inverse function, we get back the original input.

Practice with our Evaluate inverse functions exercise.

Practice with our Finding inverses of linear functions exercise.