### Unit 3: Exponents and Polynomials

In this unit, you will deepen your knowledge of functions by performing algebraic operations on functions and learning how to transform graphs of functions. You will continue to define important characteristics of functions, including domain and range, after performing algebraic operations or transformations on the graphs of functions. This is another fundamental unit, so pay close attention to what kinds of transformations can be performed on the graphs of functions and how they change the key characteristics of the function's equation, domain, and range. In later units, you will apply the same transformations to the toolkit functions you discovered in Unit 2.

**Completing this unit should take you approximately 3 hours.**

Upon successful completion of this unit, you will be able to:

- perform algebraic operations on functions to construct and deconstruct composite functions;
- describe the properties of a composite function;
- identify basic graphical transformations of functions, including shifts, compressions, stretches, and reflections;
- determine whether a function is even, odd, or neither;
- verify the inverse of a function;
- define the inverse of a function and its domain and range using algebraic operations; and
- given a one-to-one function construct the graph of its inverse.

- perform algebraic operations on functions to construct and deconstruct composite functions;

### 3.1: Composite Functions

Composite functions are combinations of functions. This section will teach you how to manipulate composite functions algebraically, define their domain and range, and graph them.

In this section, you will learn how to define the domain of a composite function.

### 3.2: Transformations

In the next few sections, you will begin to strengthen your ability to graph functions without the aid of a graphing tool and without having to do a lot of algebra. You will learn some basic transformations that can be done to the graphs of the toolkit functions to make more complex functions. For example, we can take the graph of the square root function , shift it to the left or right, and determine the resulting equation. Conversely we can begin with the equation of and determine what has been done to the graph of without doing a lot of algebra.

The next transformation we will learn is reflections. You will reflect graphs of functions across the axes and determine how the transformations change the equations of the functions.

Graphs can give us important clues about the behavior and characteristics of functions. In this section, you will explore the idea of a one-to-one function. This is an important concept for defining the inverse of a function. If you move on to calculus, this is an important concept that will return when you learn about rates of change and derivatives.

The last set of transformations you will explore on the graphs of functions are stretches and compressions. You will find out how the equation of a function changes when you stretch the graph in the x-direction or the y-direction.

### 3.3: Inverse Functions

Inverse functions bring together the concepts we have learned about composite functions and domain and range, so make sure you are confident with those sections before you dig in. A function must be one-to-one to have an inverse. If you are uncertain how to determine whether a function is one-to-one, it may be a good idea to revisit that concept.