27 by jizhen1947

VIEWS: 0 PAGES: 16

									          2.7
Function Operations and
Composition
         Operations on Functions
• Given two functions f and g, then for all values of
  x for which both f(x) and g(x) are defined, the
  functions f + g, f - g, fg, and are defined as
  follows:

   – (f + g)(x) = f(x) + g(x)   Sum
   – (f - g)(x) = f(x) - g(x)   Difference
   – (fg)(x) = f(x) × g(x)      Product

   –                            Quotient
                      Example
Let f(x) = x + 1 and g(x) = 2x2 - 3.

•   Find (f + g)(3)

•   Find (fg)(-2)
            Example continued
Solution:
• Since f(3) = 4 and g(3) = 15,
 (f + g)(3) = f(3) + g(3)  (f + g)(x) = f(x) + g(x)
           = 4 + 15
           = 19
• Since f(-2) = -1 and g(-2) = 5,
 (fg)(-2) = f(-2) × g(-2)      (fg)(x) = f(x) × g(x)
          = -1 × 5
          = -5
                         Domains
    For functions f and g, the domain of f + g, f - g, and fg
     include all real numbers in the intersection of the domains
     of f and g, while the domain of includes those real
     numbers in the intersection of the domains of f and g for
     which g(x) ¹ 0.

Example: Let f(x) = 3x - 5 and g(x) =
•   Find (f - g)(x)
•   Find

•   Give the domains of each function.
                Domains continued
Solutions:
•   (f - g)x = f(x) – g(x) =

•


•    In part (a), the domain of f is the set of all real numbers
    (- , ), and the domain of g, since
      includes just the real numbers to make 3x - 2 nonnegative.
     That is          , so     .
         Domains continued
The domain of g is      . The domain of f - g is
the intersection of the domains of f and g,
which is                  .

The domain of includes those real numbers
in the intersection above for which
; that is, the domain of is     .
       The Difference Quotient
 Suppose the point P lies on the graph of y =
 f(x), and h is a positive number. If we let (x,
 f(x)) denote the coordinates of P and (x + h, f(x
 + h)) denote the coordinates of Q, then the
 line joining P and Q has slope




This expression is called the difference quotient.
                             Example
  Let f(x) = x2 + 4x. Find the difference quotient and simplify the
   expression.

Solution:
Step 1     Find f(x + h)
           Replace x in f(x) with x + h.
           f(x + h) = (x + h)2 + 4(x + h)
Step 2 Find f(x + h) - f(x)
  f(x + h) - f(x) = [ (x + h)2 + 4(x + h)] - (x2 + 4x) Substitute.
                = x2 + 2xh + h2 + 4(x + h) - (x2 + 4x) Square x + h.
                = x2 + 2xh + h2 + 4x + 4h - x2 - 4x
                = 2xh + h2 + 4h
           Example continued
Step 3 Find the difference quotient.
     Composition of Functions
If f and g are functions, then the composite
functions, or composition, of g and f is defined
by




The domain of       is the set of all numbers x
in the domain of f such that f(x) is in the
domain of g.
                    Example
Let f(x) = 3x - 1 and g(x) = x + 5. Find each
composition.
•

Solution:
First find g(2). Since g(x) = x + 5,
                     g(2) = 2 + 5 = 7.
Now find             = f[g(2)] = f(7):
                      f[g(2)] = f(7) = 3(7) - 1 = 20.
            Example continued
•

Solution:
First find f(2). Since f(x) = 3x - 1
                     f(2) = (3(2) - 1) = 5
Now find              = g[f(2)] = g(5)
                       g[f(2)] = g(5) = 5 + 5 = 10.
      Finding Composite Functions
• Example: Let f(x) = x2 and g(x) = x - 2.
         Find             and            .

• Solution:
          Composite Functions
• The previous example shows it is not always
  true that         .
  – In fact, the composite functions are equal only
    for a special class of functions.
   Finding Functions That Form a
         Given Composite
Example: Find functions f and g such that

Solution: Note the repeated quantity x2 + 3.
         If we choose g(x) = x2 + 3 and
         f(x) = x3 - 2x + 7 then

								
To top