Function Basics for PreCalculus
A function is a special type of rule or relationship. The difference between a function and a relation is that a function has exactly one output value (from the range) for every input value (from the domain).
Functions are very useful when you’re describing trends in business, heights of objects shot from a cannon, times required to complete a task, and so on. Functions have some special properties and operations that allow for investigation into what happens when you change the rule.
In precalculus, you’ll work with functions and function operations in the following ways:

Writing and using function notation

Determining the domain and range of different types of functions

Recognizing even and odd functions

Checking on whether a function is onetoone

Finding inverses of onetoone functions

Performing the basic operations on functions and function rules

Working with the composition of functions and the difference quotient
Don’t let common mistakes trip you up; keep in mind that when working with functions, your challenges will include

Following the order of operations when evaluating functions

Determining which values need to be excluded from a function’s domain

Working with negative signs correctly when checking for even and odd functions

Being sure a function is onetoone before trying to determine an inverse

Correctly applying function rules when performing function composition

Raising binomials to higher powers and including all the terms
Practice problems

Find the domain and range for the function.
Credit: Illustration by Thomson DigitalAnswer: domain: –5 < x; range:
or y < –1.
The domain is the set of x values, and the range is the set of y values for which the function is defined. In this case, x is defined for all real numbers greater than −5. You don’t include −5 because there’s an open dot on −5 in the graph. If
then the range is
If x > 1, then the range is all real numbers less than −1. So the range is
or y < –1.

Find the inverse of the function:
Answer:
Change f(x) to y:
Interchange x and y:
Now solve for y:
Rename y as f^{–1}(x):