Functions and Graphs

Chapter 1.2

2

Functions

DEFINITION:A function from a set to a set is a pairing of elements from and from as ordered pair such that, for every in there is one and only one in .

We commonly refer to as the independent quantity and to as the dependent quantity.

The set from which the independent quantity is taken is called the domain of the function

The set which includes the dependent quantity is called the range of the function

3

Example 1: The Circle-Area Function

Write a formula that expresses the area, , of a circle as a function of its radius . Use the formula to find the area of a circle of radius 2 inches.

4

Example 1: The Circle-Area Function

Write a formula that expresses the area, , of a circle as a function of its radius . Use the formula to find the area of a circle of radius 2 inches.

The familiar area formula, can be expressed in function notation as

Given a radius of 2 inches, the area is

5

Viewing and Interpreting Graphs

DEFINITION:The points in the plane whose coordinates are the number pairs of a function make up the function’s graph.

6

Domain and Range

• Your textbook asks you to find the domain and range of a function by viewing the graph of the function

• Although the graph of a function can give you an idea of the domain and range, it is important to find these values analytically

• In the previous example (), the domain is determined by context (radius cannot be negative) so the domain is

• If the domain of a function is not stated, it is assumed that the domain is the largest set of -values for which the equation gives real -values

7

Domain and Range

• In most cases, the domain and range of a function can be expressed as intervals or as inequalities (either is acceptable)

• The figure on the right is from page 13 of your textbook

8

Example 2: Identifying Domain and Range of a FunctionIdentify the domain and range, and then sketch a graph of the function.

a)

b)

9

Example 2: Identifying Domain and Range of a FunctionIdentify the domain and range, and then sketch a graph of the function.

a) Since division by zero is undefined, may take any value but zero. The domain can be expressed as .

We can find the range by solving for and determining the values of that give a value for (though we must be careful with this method, as you will see in b). Since , then can take any non-zero value so the range is also

10

Example 2: Identifying Domain and Range of a Function

11

Example 2: Identifying Domain and Range of a FunctionIdentify the domain and range, and then sketch a graph of the function.

b) Since will only produce real values for a positive radicand, then the domain is

Now, we also have and since , then . So the range is

12

Example 2: Identifying Domain and Range of a Function

13

Example 3: Identifying Domain and Range of a FunctionIdentify the domain and range, then draw a graph of the function.

a)

b)

14

Example 3: Identifying Domain and Range of a FunctionIdentify the domain and range, then draw a graph of the function.

a) To find the domain, the radicand must not be negative. So we have

The domain is therefore or .

For the range, since , then . But there must be an upper bound on since, for example, if , then . If we solve for we get . Therefore, . Since we also have , then the range is , or . This is confirmed by the graph (next slide)

15

Example 3: Identifying Domain and Range of a Function

16

Example 3: Identifying Domain and Range of a FunctionIdentify the domain and range, then draw a graph of the function.

b) This can also be written as . Since the cube root function has no restrictions, then the domain is .

Note that, because of , the values of cannot be negative. So the range is

This is confirmed by the graph (next slide)

17

Example 3: Identifying Domain and Range of a Function

18

Even and Odd Functions—Symmetry

DEFINITION:A function is

an even function of if

an odd function of if

for every in the domain of the function.

19

Even and Odd Functions—Symmetry

The graph of an even function is symmetric about the -axis; that is, for every point , there is a point (think about )

20

Even and Odd Functions—Symmetry

The graph of an odd function is symmetric about the origin; that is, for every point , there is a point (think about )

21

Example 4: Recognizing Even & Odd FunctionsDetermine whether the functions below are even, odd, or neither:

a) b)

c) d)

22

Example 4: Recognizing Even & Odd FunctionsDetermine whether the functions below are even, odd, or neither:

a) b)

c) d) Use the definitions! For a), , even

b) , even

c) ; odd

d) and ; neither

Note that knowing the symmetry of a function can be using when graphing

Example 5: Writing Formulas for Piecewise FunctionsGraph

23

Example 5: Writing Formulas for Piecewise FunctionsGraph

Although there are three separate functions, they are part of one function that changes at and . Note that the domain is and the range is . You must choose which of the three parts to use when evaluating. For example, to evaluate use since 0.1 falls between 0 and 1.

24

Example 5: Writing Formulas for Piecewise Functions

25

Example 6: Writing Formulas for Piecewise FunctionsWrite a formula for the function whose graph consists of the line segments shown below.

26

Example 6: Writing Formulas for Piecewise FunctionsWrite a formula for the function whose graph consists of the line segments shown below.The first segment has equation with . Note that 1 is not part of this piece. The second has equation with . Hence,

The domain of the function is and the range is .

27

The Absolute Value Function

The absolute value function, can be defined piecewise as

It is generally better to use three pieces so as to avoid confusion about which piece should include zero. Later we will find that it will be easier to work with absolute value functions by breaking them into pieces. For example, the function can be written as

The graph is shown below.

28

The Absolute Value Function

29

Example 7: Using Transformations

Draw the graph of , then find its domain and range.

30

Example 7: Using Transformations

Draw the graph of , then find its domain and range.The vertex occurs at , so if , then (positive) and the graph is that part of the line that lies to the right of .

If , then or and the graph is that part of the line that lies to the left of . The vertex is at .

The domain of the absolute value function is . This function opens up (why?), so all function values are greater than or equal to . So the range is or .

31

Example 7: Using Transformations

32

Composite Functions

• It is important to keep in mind that in any function , y (and hence also ) is a real number.

• Just as we may input any value of x from the domain to evaluate a function, we may also use another function as input

• If we have two functions of x, f and g, we can compose the functions as

• Here, the function g is an input for the function f

• A common notation for composition of functions is

• This is read as “f of g of x”

33

Example 8: Composing Functions

Find a formula for if and . Then find .

34

Example 8: Composing Functions

Find a formula for if and . Then find .

Note that we could also evaluate this without the explicit equation

35

Composite Functions

• A word about the domain of composite functions (this is not in your text):• Let be functions of • The domain of is the conjunction (or intersection) of the domain of with all

values in the range of that are also in the domain of • To find the domain of this composition, you would first find the domain of ,

then find and exclude from the domain of the composition all values of that produce a value in that are not in the domain of • An example follows

36

Composite Functions

Let and . Find the domain of the composition .

37

Composite Functions

Let and . Find the domain of the composition .Since , then the domain for is . The domain for is . Therefore, we must eliminate all values of in that produce 1:

So the domain of is the intersection of

38

Composite Functions

39

Exercise 1.2

Page 19, #1-4, 5-29 odds, 41-48, 67-70 (29 total)

40