### Learning Outcomes

By the end of this section, you will be able to:

- Find the domain of a square root and rational function
- Find the domain of a function from its algebraic form.
- Define the domain of linear, quadratic, radical, and rational functions from graphs

You’ve seen that algebraic expressions such as polynomials and rational expressions can be manipulated arithmetically. That is, we can perform the operations of arithmetic on them by adding, subtracting, multiplying, and dividing expressions. When we do these arithmetic operations on algebraic expressions, we call them *algebraic operations*.

In this module, you’ll learn how to manipulate functions by performing algebraic operations to *combine* them arithmetically. You’ll also see a new way to combine functions called *composition of function*, in which a new function is created by using the output of one function as the input for another function.

Functions can be manipulated in graphical form, too. We can shift them up or down, left or right on the coordinate plane, stretch or shrink them, or flip them over vertically or horizontally. And we can perform two or more of these transformations at once. We’ll see how these *transformations* of the graphs of functions appear in the plane and how they can be detected in the equation of a function.

Finally, we’ll look at *inverse* functions. You are already familiar with pairs of inverse operations. We think of subtraction for instance as reversing the procedure of addition and division as the “opposite” operation of multiplication. Cubing the cube root of a number undoes both operations and returns to us just the number inside the radical. The cube root function is thus said to be the *inverse* of the cubing function. We’ll examine inverse operations algebraically and graphically.

Each time we combine functions, we create a new function that must be carefully defined with regard to what values may be used as input to the function. Before we learn how to combine functions, it will be necessary to formally discuss an idea you’ve seen already regarding rational and radical expressions: **domain restrictions**.

When we transform functions, the domain or range can become altered. So we’ll also take a look in this review section at **reading the domain and range of a function from a graph**.

Warm up for this module by refreshing important concepts and skills you’ll need for success. As you study these review topics, recall that you can also return to Algebra Essentials any time you need to refresh the basics.

### Recall for success

Look for red boxes like this one throughout the text. They’ll show up just in time to give helpful reminders of the math you’ll need, right where you’ll need it.