chapter 1: prerequisites for calculus section 1.2: functions and graphs ap calculus ab

CHAPTER 1:PREREQUISITES FOR CALCULUS

SECTION 1.2:FUNCTIONS AND GRAPHS

AP CALCULUS AB

FunctionsDomains and RangesViewing and Interpreting GraphsEven Functions and Odd functions -

SymmetryFunctions Defined in PiecesAbsolute Value FunctionComposite Functions

…and why

Functions and graphs form the basis for understanding mathematics applications.

What you’ll learn about…

Functions

A rule that assigns to each element in one set a unique element in another set is called a function. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range.

Function

A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

In this definition, D is the domain of the function and R is a set containing the range.

Function

( )

( )( )

The symbolic way to say " is a function of " is which is read as equals of .The notation gives a way to denote specific values of a function. The value of at can be written as , read

y x y f xy f x

f xf a f a

=

as " of ."f a

Example Functions

Evaluate the function ( ) 2 3 when = 6. f x x x= +

( )

( ) 2( ) 3

( ) 12 3

15

6 6

6

6

f

f

f

= +

= +

=

Domains and Ranges

( )When we define a function with a formula and the domain is

not stated explicitly or restricted by context, the domain is assumed to be

the largest set of -values for which the formula gives real

y f x

x

=

( )

-values -

the so-called natural domain. If we want to restrict the domain, we must say so.

The domain of 2 is restricted by context because the radius, ,

must always be positive.

y

C r r rp=

Domains and Ranges

The domain of 5 is assumed to be the entire set of real numbers.

If we want to restrict the domain of 5 to be only positive values,

we must write 5 , 0.

y x

y x

y x x

=

=

= >

Domains and Ranges

The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed or half-open, finite or infinite.

The endpoints of an interval make up the interval’s boundary and are called boundary points.

The remaining points make up the interval’s interior and are called interior points.

Domains and Ranges

Closed intervals contain their boundary points.Open intervals contain no boundary points

Domains and Ranges

Graph

( )( )

The points , in the plane whose coordinates are the

input-output pairs of a function make up the

function's .

x y

y f x

graph

=

Example Finding Domains and Ranges

2

Identify the domain and range and use a grapher

to graph the function .y x=

( )

[ )

Domain: The function gives a real value of for every value of

so the domain is , .

Range: Every value of the domain, , gives a real, positive value of

so the range is 0, .

y x

x y

- ¥ ¥

¥

[-10, 10] by [-5, 15]

2y x=

Viewing and Interpreting Graphs

Recognize that the graph is reasonable.

See all the important characteristics of the graph.

Interpret those characteristics.

Recognize grapher failure. (Occurs when the graph produced by your calculator is less than precise, or even incorrect, usually due to the limitations of the screen resolution of the grapher.)

Graphing with a graphing calculator requires that you develop graph viewing skills.

Viewing and Interpreting Graphs

Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs.

Grapher failure occurs when the graph produced by a grapher is less than precise – or even incorrect – usually due to the limitations of the screen resolution of the grapher.

Example Viewing and Interpreting Graphs

2

Identify the domain and range and use a grapher to

graph the function 4y x= -

( ] [ )

Domain: The function gives a real value of for each value of 2

so the domain is , 2 2, .

Range: Every value of the domain, ,

gives a real, positive value of

so the range is [0, ).

y x

x

y

³

- ¥ - È ¥

¥

[-10, 10] by [-10, 10]

2 4y x= -

Section 1.2 – Functions and Graphs

Example: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

2

: 5,5

: 0,5

25

y

D

R

x

Section 1.2 – Functions and Graphs

Example: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

1 3

: - ,

: - ,

D

R

y x

Section 1.2 – Functions and Graphs

You try: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

248

y x

Section 1.2 – Functions and Graphs

You try: Use a graphing calculator to identify the domain and range, then draw the graph of the function.

1 23

y x

Even Functions and Odd Functions-Symmetry

The graphs of even and odd functions have important symmetry properties.

( )( ) ( )

A function ( )is a

if ( )

if

for every in the function's domain.

y f x

x f x f x

x f x f x

x

=

- =

- =-

even function of

odd function of

Even Functions and Odd Functions-Symmetry

The graph of an even function is symmetric about the y-axis. A point (x,y) lies on the graph if and only if the point (-x,y) lies on the graph.(Plug in –x for x and see if it yields the same equation).

The graph of an odd function is symmetric about the origin. A point (x,y) lies on the graph if and only if the point (-x,-y) lies on the graph.(Plug in –x for x and –y for y and see if it yields the same equation.)

Example Even Functions and Odd Functions-Symmetry

3Determine whether is even, odd or neither.y x x= -

( ) ( ) ( ) ( ) ( )

3

3 3 3

is odd because

x

y x x

f x xx x f xx x- -

= -

= - = - + = - - =--

3y x x= -

Example Even Functions and Odd Functions-Symmetry

Determine whether 2 5 is even, odd or neither.y x= +

( ) ( ) ( )2 5 is neither because

2 5 2 5 ( )x x

y x

f x f x f x

= +

= + =- + ¹ ¹ -- -

2 5y x= +

Section 1.2 – Functions and Gr

More Examples:

4

4 4Even function: for all

Symmetric about the -axis

Odd function: for all

Symmetric about the o

1.

2.

3. 2

rigin

Not odd: 2 2

f x x x f x x

y

f x x x f x x

f x x

f x x

f

x x

x

f x

x

Not even: 2 2

f x

f x x x f x

Section 1.2 – Functions and Gr

You try: Determine whether the following functions are even, odd or neither.

3

2

1. 5

22.

9

f x x

f xx

Functions Defined in Pieces

While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain.

These are called piecewise functions.

Example Graphing a Piecewise Defined Function

2

Use a grapher to graph the following piecewise function :

2 1 0( )

3 0

x xf x

x x

2 1; 0y x x= - £

2 3; 0y x x= + >

[-10, 10] by [-10, 10]

Section 1.2 – Functions and Graphs

Example 2:

2 1, 0( )

1, 0

x xf x

x x

Absolute Value Functions

Absolute Value Function can be defined as a piecewise function.

, 0

, 0

x xx

x x

The function is even, and its graph is symmetric about the y-axis

Section 1.2 – Functions and GraphsExample: Write a formula for the piecewise

function whose graph consists of 2 segments, one from (0, 0) to (1, 2) and the other from (1, 0) to (2, 2)

Segment from (0, 0) to (1, 2)

Segment from (1, 0) to (2, 2)

2 02

2 10 2 1

2 02, 0

1

2

0

2

m

x

m

x

y

b

y

2 , 0 1,

2 2, 1 2

x xSo f x

x x

Section 1.2 – Functions and GraphsYou try: Write a formula for the piecewise function

whose graph consists of 2 segments, one from (-4, -1) to (-2, 0) and the other from (-2, -1) to (0, 0)

Composite Functions

Suppose that some of the outputs of a function can be used as inputs of

a function . We can then link and to form a new function whose inputs

are inputs of and whose outputs are the numbers

g

f g f

x g ( )( )( )( ) ( )

.

We say that the function read of of is

. The usual standard notation for the composite is ,

which is read " of ."

f g x

f g x f g x

f g

f g

the composite

of and og f

Example Composite Functions

( )Given ( ) 2 3 and 5 , find .f x x g x x f g= - = o

( ) ( )( )( )( )

( )

2 3

1

5

5

0 3

g x

x

f g x

f

x

x

f=

=

= -

= -

o

In Review

Bounded Intervals:1. [a, b] means

it is a closed interval

2. (a, b) means it is an open interval

3. [a, b) means

4. (a, b] means

bxa

bxa

bxa

bxa

[ ]a b

( )a b

[ )a b

( ]a b

In Review

Unbounded Intervals1. means

2. means

3. means

4. means

5. means the entire real number line.

),[ a

),( a

],( b

),( b

),(

ax

ax

bx

bx

[a

(a

]b

)b

In Review

A function from a set D (domain) to a set R (range) is a rule that assigns a unique element in R to each element in D.

The domain is the largest set of x-values for which the formula gives real y-values.

The range is the set of y-values yielded by the function.

In Review

Other symmetries:Symmetric about the x-axis: Plug in –y for y and see if it yields the same equation.

Symmetric about the y = x line: Switch the x’s and y’s and see if it yields the same equation.

In ReviewComposite Functions – work from the inside out

Ex:

2

2

2

22

and

( )

2 3

(

) 2 3

2 3

2

( )

2

3 3

f g x g f

f x x

x

g x x

gf f x

f x

x

x

x

g

x g

x

2 4 12 9x x

Section 1.2 – Functions and Graphs

You try: Find a formula for f(g(x)) for the following pair of functions, then find

f(g(-2)) .

21. 3 2 and 4g x x f x x