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Evaluation and Composition of function

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					             Evaluation and composition of function
    Evaluation:
•   The letter f in the function notation f(x) as the name of the function. Instead of
    using the equation                 to describe the function,

•   we can write                   . Here, f is the name of the function and f(x) is the
    value of the function at x. So                                  is the value of the
    function at 2.
•   The notation affords a convenient way of prompting the evaluation of a function
    for a particular value of x.
•   Any letter can be used as the independent variable in a function. So the above
    function could be written                    .

•   The independent variable in a function is just a place holder. The function could
    be written without a variable as follows




•   The function can be viewed as an input/output operation.
•   In addition to plugging numbers into functions, we can plug expressions into
    functions. Plugging y + 1 into the function                    yields




•   Plug other expressions in terms of x into a function. Plugging 2x into the function

                                   yield




•   The variable x in the function is being replaced by the same variable. But the x in
    function is just a placeholder. If the placeholder were removed from the function,
    the substitution would appear more natural. In                      , we plug 2x into
    the left side f(2x) and it returns the right side
Composition

 •   We have plugged numbers into functions and expressions into functions; now
     let’s plug in other functions. Since a function is identified with its expression.
 •   In the example above with                     and 2x, let’s call 2x by the name g(x).
     In other words, g(x) = 2x. Then the composition of f with g (that is plugging g into
     f) is




 •   The notation f(g(x)) on the test. But you probably will see one or more problems

     that ask you perform the substitution. For another example, let                  and

     let             . Then




                           and


 •   The composition of functions merely substitutes one function into another, these
     problems can become routine. Notice that the composition operation f(g(x)) is
     performed from the inner parentheses out, not from left to right. In the operation
     f(g(2)), the number 2 is first plugged into the function g and then that result is
     plugged in the function f.
 •   A function can also be composed with itself. That is, substituted into itself. Let
                      . Then
Example:1

Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(x).

   A) –2x2 + 13

   B) 2x2 + 13

   C) –2x2 – 13

   D) 2x2 – 13

   Solution:

   plugging the formula for g(x) into the formula for f(x)

       ( f o g)(x) = f (g(x))
           = f (–x2 + 5)
           = 2(            )+3   ... setting up to insert the input formula
           = 2(–x2 + 5) + 3
           = –2x2 + 10 + 3
           = –2x2 + 13

Example:2

Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(1)

 (A) 11

 (B) -11

 (C) 12

 (D) -12

 (E) 13

 Solution:
( f o g)(1) = f (g(1))
   = f (–(    )2 + 5)    ... setting up to insert the original input
   = f (–(1)2 + 5)
   = f (–1 + 5)
   = f (4)
   = 2( ) + 3        ... setting up to insert the new input
   = 2(4) + 3
   =8+3
   = 11

Practice Questions

  1) The graph of y = f(x) is shown to the right. If f(-1) = v, then which one of the
      following could be the value of f(v) ?




      (A) 0
      (B) 1
      (C) 2
      (D) 2.5
      (E) 3
 2) In the function above, if f(k) = 2, then which one of the following could be a value
      of k ?
      (A) -1
      (B) 0
      (C) 0.5
      (D) 2.5
      (E) 4
  3) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(x).
      (A) 4x2 + 12x – 4
      (B) –4x2 – 12x – 4
      (C) –4x2 – 12x + 4
      (D) 4x2 + 12x + 4
      (E) –4x2 + 12x + 4
  4) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o f )(x)
      (A) 4x – 9
      (B) 4x + 7
      (C) 4x + 9
      (D) -4x + 9
      (E) -4x - 9
  5) Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(1)
      (A) 20
      (B) 18
      (C) 19
      (D) -20
      (E) 21




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DOCUMENT INFO
Description: Free SAT worksheet on evaluation and composition of functions. The worksheet includes explanation, solved example problems and practice problems