chapter 1: prerequisites for calculus section 1.2: functions and graphs ap calculus ab
TRANSCRIPT
![Page 1: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/1.jpg)
CHAPTER 1:PREREQUISITES FOR CALCULUS
SECTION 1.2:FUNCTIONS AND GRAPHS
AP CALCULUS AB
![Page 2: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/2.jpg)
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…
![Page 3: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/3.jpg)
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.
![Page 4: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/4.jpg)
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.
![Page 5: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/5.jpg)
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
![Page 6: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/6.jpg)
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
= +
= +
=
![Page 7: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/7.jpg)
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=
![Page 8: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/8.jpg)
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
=
=
= >
![Page 9: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/9.jpg)
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.
![Page 10: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/10.jpg)
Domains and Ranges
Closed intervals contain their boundary points.Open intervals contain no boundary points
![Page 11: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/11.jpg)
Domains and Ranges
![Page 12: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/12.jpg)
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
=
![Page 13: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/13.jpg)
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=
![Page 14: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/14.jpg)
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.
![Page 15: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/15.jpg)
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.
![Page 16: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/16.jpg)
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= -
![Page 17: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/17.jpg)
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
![Page 18: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/18.jpg)
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
![Page 19: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/19.jpg)
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
![Page 20: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/20.jpg)
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
![Page 21: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/21.jpg)
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
![Page 22: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/22.jpg)
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.)
![Page 23: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/23.jpg)
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= -
![Page 24: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/24.jpg)
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= +
![Page 25: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/25.jpg)
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
![Page 26: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/26.jpg)
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
![Page 27: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/27.jpg)
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.
![Page 28: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/28.jpg)
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]
![Page 29: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/29.jpg)
Section 1.2 – Functions and Graphs
Example 2:
2 1, 0( )
1, 0
x xf x
x x
![Page 30: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/30.jpg)
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
![Page 31: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/31.jpg)
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
![Page 32: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/32.jpg)
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)
![Page 33: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/33.jpg)
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
![Page 34: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/34.jpg)
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
![Page 35: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/35.jpg)
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
![Page 36: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/36.jpg)
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
![Page 37: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/37.jpg)
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.
![Page 38: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/38.jpg)
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.
![Page 39: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/39.jpg)
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
![Page 40: CHAPTER 1: PREREQUISITES FOR CALCULUS SECTION 1.2: FUNCTIONS AND GRAPHS AP CALCULUS AB](https://reader033.vdocuments.mx/reader033/viewer/2022061614/56649ea25503460f94ba5cbc/html5/thumbnails/40.jpg)
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