AP CALCULUS AB
CHAPTER 1:
PREREQUISITES FOR CALCULUS
SECTION 1.2:
FUNCTIONS AND GRAPHS
What you’ll learn about…
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Functions
Domains and Ranges
Viewing and Interpreting Graphs
Even Functions and Odd functions - Symmetry
Functions Defined in Pieces
Absolute Value Function
Composite Functions
…and why
Functions and graphs form the basis for
understanding mathematics applications.
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 " y is a function of x " is y = f (x )
which is read as y equals f of x.
The notation f (x) gives a way to denote specific values of a function.
The value of f at a can be written as f (a ), read as " f of a."
Example Functions
Evaluate the function f ( x) = 2 x + 3 when x = 6.
f (6) = 2(6) + 3
f (6) = 12 + 3
f (6)= 15
Domains and Ranges
When we define a function y = f (x) with a formula and the domain is
not stated explicitly or restricted by context, the domain is assumed to be
the largest set of x-values for which the formula gives real y -values the so-called natural domain. If we want to restrict the domain, we must say so.
The domain of C (r )= 2p r is restricted by context because the radius, r ,
must always be positive.
Domains and Ranges
The domain of y = 5 x is assumed to be the entire set of real numbers.
If we want to restrict the domain of y = 5 x to be only positive values,
we must write y = 5 x, x > 0.
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 (x, y )in the plane whose coordinates are the
input-output pairs of a function y = f (x)make up the
function's graph.
Example Finding Domains and
Ranges
Identify the domain and range and use a grapher
to graph the function y = x 2 .
Domain: The function gives a real value of y for every value of x
so the domain is (- ¥ , ¥ ).
Range: Every value of the domain, x, gives a real, positive value of y
so the range is [0, ¥ ).
y = x2
[-10, 10] by [-5, 15]
Viewing and Interpreting Graphs
Graphing with a graphing calculator requires that you
develop graph viewing skills.
 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.)
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
Identify the domain and range and use a grapher to
graph the function y =
x2 - 4
Domain: The function gives a real value of y for each value of x ³ 2
so the domain is (- ¥ , - 2]È [2, ¥ ).
Range: Every value of the domain, x,
gives a real, positive value of y
so the range is [ 0, ¥ ).
y=
x2 - 4
[-10, 10] by [-10, 10]
Section 1.2 – Functions and Graphs
 Example: Use a graphing calculator to identify the
domain and range, then draw the graph of the
function.
y  25  x 2
D :  5,5
R:
0,5
Section 1.2 – Functions and Graphs
 Example: Use a graphing calculator to identify the
domain and range, then draw the graph of the
function.
y  x1 3
D :  -, 
R:
 -, 
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.
y  48  x 2
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.
y  3 x1 2
Even Functions and
Odd Functions-Symmetry
 The graphs of even and odd functions have important
symmetry properties.
A function y = f ( x)is a
even function of x if f (- x) = f (x )
odd function of x if f (- x )= - f (x )
for every x in the function's domain.
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
Determine whether y = x 3 - x is even, odd or neither.
y = x3 - x is odd because
3
f (- x)= (- x) - (- x) = - x3 + x = - (x3 - x)= - f (x)
y = x3 - x
Example Even Functions and Odd
Functions-Symmetry
Determine whether y = 2 x + 5 is even, odd or neither.
y = 2 x + 5 is neither because
f (- x )= 2 (- x )+ 5 = - 2 x + 5 ¹ f ( x) ¹ - f (x )
y = 2x + 5
Section 1.2 – Functions and Gr
 More Examples:
1. f  x    x 4
Even function: f   x      x    x 4  f  x  for all x
4
Symmetric about the y -axis
2. f  x    x
Odd function: f   x      x   x   f  x  for all x
Symmetric about the origin
3. f  x    x  2
Not odd: f   x      x   2  x  2   f  x 
Not even: f   x      x   2  x  2  f  x 
Section 1.2 – Functions and Gr
 You try: Determine whether the following functions are
even, odd or neither.
1. f  x   5 x 3
2. f  x  
2
x2  9
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
Use a grapher to graph the following piecewise function :
2 x  1 x  0
f ( x)   2
x  3 x  0
y = x 2 + 3; x > 0
y = 2 x - 1; x £ 0
[-10, 10] by [-10, 10]
Section 1.2 – Functions and Graphs
 Example 2:
 x 2  1,
f ( x)  
 x  1,
x0
x0
Absolute Value Functions
 Absolute Value Function can be defined as a piecewise
function.
  x,
x 
 x,
x0
x0
The function is even, and its graph is symmetric about the y-axis
Section 1.2 – Functions and Graphs
 Example: 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)
20
m
 2, b   0
1 0
Segment from (1, 0) to (2, 2)
20
m
2
2 1
y  0  2  x  1
y  2x  2
0  x 1
2 x,
So, f  x   
2 x  2, 1  x  2
Section 1.2 – Functions and Graphs
 You 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 g can be used as inputs of
a function f . We can then link g and f to form a new function whose inputs
x are inputs of g and whose outputs are the numbers f (g (x )).
We say that the function f (g (x )) read ( f of g of x )is the composite
of g and f . The usual standard notation for the composite is f o g ,
which is read " f of g ."
Example Composite Functions
Given f ( x) = 2 x - 3 and g (x ) = 5 x, find f o g.
( f o g ) (x ) = f (g (x ))
= f (5 x )
= 2 (5 x )- 3
= 10 x - 3
In Review
Bounded Intervals:
1.
[a, b] means a 
it is a closed interval
2.
(a, b) means
xb
a xb
it is an open interval
3.
4.
[a, b) means
(a, b] means
a xb
a xb
[
a
]
b
(
a
[
a
(
a
)
b
)
b
]
b
In Review
Unbounded Intervals
1.
[a, ) means
xa
2.
( a,  )
means
xa
3.
(, b]
means
xb
4.
(, b)
means
xb
5.
(, ) means the entire
real number line.
[
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 Review
 Composite Functions – work from the inside out
Ex:
f ( x)  2 x  3 and g ( x)  x 2
f
g  ( x)  f  g  x  
 
 2 x   3
 f x2
2
 2 x2  3
g
f  x   g  f  x  
 g  2 x  3
  2 x  3
2
 4 x 2  12 x  9
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)) .
1. g  x   3x  2 and f  x   4 x2