Thinking Mathematically by Robert Blitzer by yurtgc548

VIEWS: 2 PAGES: 19

									Graphs of
Functions
                           Text Example
Graph f (x) = x2 + 1. To do so, use integer values of x
from the set {-3, -2, -1, 0, 1, 2, 3} to obtain seven
ordered pairs. Plot each ordered pair and draw a smooth
curve through the points. Use the graph to specify the
function's domain and range.
Solution The graph of f (x) = x2 + 1 is, by definition, the graph of y = x2 +
1.
We begin by setting up a partial table of coordinates.
                     2
           x   f (x) = x + 1             (x, y) or (x, f (x))
          -3   f (-3) = (-3)2 + 1 = 10   (-3, 10)
          -2   f (-2) = (-2)2 + 1 = 5    (-2, 5)
          -1   f (-1) = (-1)2 + 1 = 2    (-1, 2)
           0   f (0) = 02 + 1 = 1        (0, 1)
                      Text Example cont.
Solution              x     f (x) = x2 + 1              (x, y) or (x, f (x))
                      1     f (1) = 12 + 1 = 2          (1, 2)
                      2     f (2) = (2)2 + 1 = 5        (2, 5)
                      3     f (3) = (3)2 + 1 = 10       (3, 10)


Now, we plot the seven points and draw a smooth curve                                               10

through them, as shown. The graph of f has a cuplike shape.                                          9
                                                                                                     8
The points on the graph of f have x-coordinates that extend                                          7




                                                                        Range: [1, oo)
                                                                                                     6
indefinitely far to the left and to the right. Thus, the domain                                      5
consists of all real numbers, represented by (-oo, oo). By                                           4
                                                                                                     3
contrast, the points on the graph have y-coordinates that start at                                   2

1 and extend indefinitely upward. Thus, the range consists of all                                    1


real numbers greater than or equal to 1, represented by [1, oo).                         -4 -3 -2 -1      1   2    3   4

                                                                                               Domain: all Reals
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.
                            Text Example
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)?                                          5
                                                                        4


• What is the range of f (x)?                                           3
                                                                        2
                                                                        1

                                                        -5 -4 -3 -2   -1     1   2   3 4   5
Solution                                                                -1
                                                                        -2

a. Because (-1, 2) is a point on the graph of f, the                   -3
                                                                       -4

   y-coordinate, 2, is the value of the function at                    -5



   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.
                    Text Example cont.
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)?                                            5
                                                                          4


• What is the range of f (x)?                                             3
                                                                          2
                                                                          1

                                                          -5 -4 -3 -2   -1     1   2   3 4   5
Solution                                                                  -1
                                                                          -2

b. The open dot on the left shows that x = -3 is not in                  -3
                                                                         -4

   the domain of f. By contrast, the closed dot on the                   -5



   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 < x < 6} or the interval (-3, 6].
                    Text Example cont.
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)?                                           5
                                                                         4


• What is the range of f (x)?                                            3
                                                                         2
                                                                         1

                                                         -5 -4 -3 -2   -1     1   2   3 4   5
Solution                                                                 -1
                                                                         -2

c. The points on the graph all have y-coordinates                       -3
                                                                        -4

   between -4, not including -4, and 4, including 4.                    -5



   The graph does not extend below y = -4 or above
   y = 4. Thus, the range of f is { y | -4 < y < 4} or
   the interval (-4, 4].
     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.
                                          Text Example
Use the vertical line test to identify graphs in which y is a function of x.
 a.           y                   b.         y                     c.         y                     d.           y




                             x                                 x                                x                               x




Solution y is a function of x for the graphs in (b) and (c).

 a.           y                   b.         y                     c.         y                     d.           y




                             x                                 x                                x                               x



      y is not a function since        y is a function of x.            y is a function of x.            y is not a function since
      2 values of y correspond                                                                           2 values of y correspond
      to an x-value.                                                                                     to an x-value.
      Increasing, Decreasing, and
          Constant Functions
A function is increasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) < f (x2).
A function is decreasing on an interval if for any x1, and x2 in the
interval, where x1 < x2, then f (x1) > f (x2).
A function is constant on an interval if for any x1, and x2 in the interval,
where x1 < x2, then f (x1) = f (x2).


                   (x2, f (x2))       (x1, f (x1))
                                                            (x1, f (x1))
                                             (x2, f (x2))
            (x1, f (x1))
                                                                    (x2, f (x2))


             Increasing            Decreasing                   Constant
           f (x1) < f (x2)        f (x1) < f (x2)            f (x1) < f (x2)
                        Text Example

Describe the increasing, decreasing, or constant behavior of each function
whose graph is shown.
                        a.                  5      b.                              5
                                            4                                      4
                                            3                                      3
                                            2
                                            1                                      1

                            -5 -4 -3 -2   -1     1   2   3 4   5   -5 -4 -3 -2   -1     1   2   3 4   5
                                            -1                                     -1
                                            -2                                     -2
                                           -3                                     -3
                                           -4                                     -4
                                           -5                                     -5


Solution
a. The function is decreasing on the interval (-oo, 0), increasing on the interval
   (0, 2), and decreasing on the interval (2, oo).
                  Text Example cont.
Describe the increasing, decreasing, or constant behavior of each function
whose graph is shown.
                        a.                  5      b.                              5
                                            4                                      4
                                            3                                      3
                                            2
                                            1                                      1

                            -5 -4 -3 -2   -1     1   2   3 4   5   -5 -4 -3 -2   -1     1   2   3 4   5
                                            -1                                     -1
                                            -2                                     -2
                                           -3                                     -3
                                           -4                                     -4
                                           -5                                     -5


Solution
b. Although the function's equations are not given, 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 on the interval (-oo, 0). The
   part to the right of the y-axis shows that the function is increasing on the
   interval (0, oo).
  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(x) > 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) < f(x) for all
   x in the open interval.
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
    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.
                    Example
Identify the following function as even, odd, or
   neither: f(x) = 3x2 - 2.
Solution:
We use the given function’s 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.
    Even Functions and y-Axis
           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.
Graphs of
Functions

								
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