Obtaining Information from Graphs

1
Obtaining Information from Graphs
You can obtain information about a function from
its graph. At the right or left of a graph, you
will find closed dots, open dots, or arrows.

• A closed dot indicates that the graph does not
  extend beyond this point and the point belongs to
  the graph.
• An open dot indicates that the graph does not
  extend beyond this point and the point does not
  belong to the graph.
• An arrow indicates that the graph extends
  indefinitely in the direction in which the arrow
  points.

2
Example Obtaining Information from a
Functions Graph

• Use the graph of the function f to answer the
  following questions.
• What are the function values f (-1) and f (1)?
• What is the domain of f (x)?
• What is the range of f (x)?

Solution a. Because (-1, 2) is a point on the
graph of f, the y-coordinate, 2, is the value of
the function at the x-coordinate, -1. Thus, f
(-l) 2. Similarly, because (1, 4) is also a
point on the graph of f, this indicates that f
(1) 4.
3
Example Obtaining Information from a
Functions Graph
Solution b. The open dot on the left shows that
x -3 is not in the domain of f. By contrast,
the closed dot on the right shows that x 6 is.
We determine the domain of f by noticing that the
points on the graph of f have x-coordinates
between -3, excluding -3, and 6, including 6.
Thus, the domain of f is x -3 lt x lt 6 or
the interval (-3, 6.
4
Example Obtaining Information from a
Functions Graph
Solution c. The points on the graph all have
y-coordinates between -4, not including -4, and
4, including 4. The graph does not extend below y
-4 or above y 4. Thus, the range of f is
y -4 lt y lt 4 or the interval (-4, 4.
5
The Vertical Line Test for Functions
If any vertical line intersects a graph in more
than one point, the graph does not define y as a
function of x.
6
Example Using the Vertical Line Test
Solution y is a function of x for the graphs in
(b) and (c).
7
Increasing, Decreasing, and Constant Functions
A function is increasing on an interval if for
any x1, and x2 in the interval, where x1 lt x2,
then f (x1) lt f (x2). A function is decreasing on
an interval if for any x1, and x2 in the
interval, where x1 lt x2, then f (x1) gt f (x2).
A function is constant on an interval if for any
x1, and x2 in the interval, where x1 lt x2, then
f (x1) f (x2).
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Example Intervals on Which a Function
Increases, Decreases, or Is Constant
Describe the increasing, decreasing, or constant
behavior of each function whose graph is shown.
Solution
a. Take note as to when the function changes
direction. The function is decreasing on the
interval (-oo, 0), increasing on the interval (0,
2), and decreasing on the interval (2, oo).
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Example Intervals on Which a Function
Increases, Decreases, or Is Constant
Describe the increasing, decreasing, or constant
behavior of each function whose graph is shown.
Solution

• The graph indicates that the function is defined
  in two pieces. The part of the graph to the left
  of the y-axis shows that the function is constant
  with an open endpoint on the right. So the
  function is constant on the interval (-oo, 0).
• The part to the right of the y-axis shows that
  the function is increasing on the interval with a
  closed dot on the left. So the function is
  increasing on the interval 0, oo).

10
Definition of Even and Odd Functions
The function f is an even function if f (-x) f
(x) for all x in the domain of f. The right side
of the equation of an even function does not
change if x is replaced with -x. The function f
is an odd function if f (-x) -f (x) for all x
in the domain of f. Every term in the right side
of the equation of an odd function changes sign
if x is replaced by -x.
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Example

• Identify the following function as even, odd, or
  neither f(x) 3x2 - 2.
• Solution
• We use the given functions equation to find
  f(-x).
• f(-x) 3(-x) 2-2 3x2 - 2.
• The right side of the equation of the given
  function did not change when we replaced x with
  -x.
• Because f(-x) f(x), f is an even function.

12
Example

• Identify the following function as even, odd, or
  neither g(x) x2 x - 2.
• Solution
• We use the given functions equation to find
  g(-x).
• g(-x) (-x) 2 (-x) - 2 x2 - x - 2.
• The right side of the equation of the given
  function changed when we replaced x with -x.
  Because g(-x) ? g(x), f is not an even function.
• Next we check to see if the function is odd where
  -g(x) g(-x).
• -g(x) -(x2 x - 2) -x2 - x 2 ? x2 - x -
  2 g(-x)
• So the function is not an odd fiunction.
  Therefore the function is neither.

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Even Functions and y-Axes Symmetry
The graph of an even function in which f (-x) f
(x) is symmetric with respect to the y-axis.
Odd Functions and Origin Symmetry
The graph of an odd function in which f (-x) -
f (x) is symmetric with respect to the origin.
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Definitions of Relative Maximum and Relative
Minimum

1. A function value f(a) is a relative maximum of f
   if there exists an open interval about a such
   that f(a) gt f(x) for all x in the open interval.
2. A function value f(b) is a relative minimum of f
   if there exists an open interval about b such
   that f(b) lt f(x) for all x in the open interval.

15
The Average Rate of Change of a Function

• Let (x1, f(x1)) and (x2, f(x2)) be distinct
  points on the graph of a function f.
• The average rate of change of f from x1 to x2 is