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Combinations of Functions Help (page 3)

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

Recognizing Complicated Functions

In order for calculus students to use some formulas, they need to recognize complicated functions as a combination of simpler functions. Sums, differences, products, and quotients are easy to see, but some compositions of functions are less obvious.

Examples

Find functions f ( x ) and g ( x ) so that h ( x ) = f o g ( x ).

  • Combinations of Functions and Inverse Functions Examples

    Although there are many possibilities for f ( x ) and g ( x ), there is usually one pair of functions that is obvious. Usually we want g ( x ) to be the computation that is done first and f ( x ), the computation to be done last. Here, when computing the y -value for h ( x ), we would calculate x + 16. This will be g ( x ). The last calculation will be to take the square root. This will be f ( x ). If we let Combinations of Functions and Inverse Functions Examples = x + 16, we have f o g ( x ) = f ( g ( x )) = Combinations of Functions and Inverse Functions Examples.

  • Combinations of Functions and Inverse Functions Examples

    When computing a y -value for h ( x ), we would first find x 2 + 1. This will be g ( x ). This number will be the denominator of a fraction whose numerator is 2. This will be f ( x ), a fraction whose numerator is 2 and whose denominator is x . If Combinations of Functions and Inverse Functions Examples and g ( x ) = x 2 + 1,

    Combinations of Functions and Inverse Functions Examples

Combinations of Functions Practice Problems

Practice

  1. f ( x ) = 3 x 2 + x and g ( x ) = x −4

    (a) Find f + g ( x ), fg ( x ), fg ( x ), Combinations of Functions and Inverse Functions Practice .

    (b) What is the domain for Combinations of Functions and Inverse Functions Practice ?

    (c) Find f o g ( x ) and g o f ( x ).

    (d) What is the domain for f o g ( x )

    (e) Find f o g (1) and g o f (0).

    (f) Find f o f ( x ).

  2. Find f o g ( x ), g o f ( x ), and the domain for f o g ( x ).

    Combinations of Functions and Inverse Functions Practice

  3. Refer to the graphs in Figure 4.2. The solid graph is the graph of f ( x ), and the dashed graph is the graph of g ( x ). Find f o g (1), f o g (4), and g o f (-2).

  4. Find f o g o h ( x ) for Combinations of Functions and Inverse Functions Practice , g ( x ) = 4 x + 9, and h ( x ) = 5 x 2 − 1.

  5. Find functions f ( x ) and g ( x ) so that h ( x ) = f o g ( x ), where h ( x ) = ( x - 5) 3 + 2.

    Combinations of Functions and Inverse Functions Practice

    Fig. 4.2 .

Solutions

  1. (a)

    Combinations of Functions and Inverse Functions Solutions

    (b) The domain is x ≠ 4, (from x − 4 = 0), or (−∞, 4) ∪ (4, ∞). (c)

    (c)

    f o g ( x ) = f ( g ( x )) = f ( x − 4) = 3( x − 4) 2 + ( x − 4)

          = 3( x − 4)( x − 4) + x − 4 = 3 x 2 − 23 x + 44

    g o f ( x ) = g ( f ( x )) = g (3 x 2 + x ) = 3 x 2 + x − 4

    (d) The domain for g ( x ) is all real numbers. We can let x be any real number for f ( x ), so we do not need to remove anything from the domain of g ( x ). The domain of f o g ( x ) is all real numbers, or (−∞, ∞).

    (e)

    f o g (1) = f ( g (1))

          = f (−3) g (1) = 1 − 4 = − 3

          = 24 f (−3) = 3(−3) 2 + (−3) = 24

    g o f (0) = g ( f (0))

          = g (0) f (0) = 3(0) 2 + 0 = 0

          = −4 g (0) = 0−4 = −4

    (f)

    f o f ( x ) = f ( f ( x )) = f (3 x 2 + x ) = 3(3 x 2 + x ) 2 + (3 x 2 + x )

          = 3(3 x 2 + x )(3 x 2 + x ) + 3 x 2 + x = 27 x 4 + 18 x 3 + 6 x 2 + x

  2.  

    Combinations of Functions and Inverse Functions Solutions

     

    Combinations of Functions and Inverse Functions Solutions

    The domain of g ( x ) is x ≠ 1. Now we need to see if there is anything we need to remove from x ≠ 1. Before simplifying f o g ( x ), we have

    Combinations of Functions and Inverse Functions Solutions

    The denominator of this fraction cannot be zero, so we must have Combinations of Functions and Inverse Functions Solutions .

    Combinations of Functions and Inverse Functions Solutions

    The domain is Combinations of Functions and Inverse Functions Solutions .

    While it seems that x = − 4 might not be allowed in the domain of f o g ( x ), x = −4 is in the domain.

    Combinations of Functions and Inverse Functions Solutions

  3.  

    f o g (1) = f ( g (1)) Look for x = 1 on g ( x ).

          = f (4) (1, 4) is on the graph of g ( x ), so g (1) = 4.

          = 1 (4, 1) is on the graph of f ( x ), so f (4) = 1.

    f o g (4) = f ( g (4)) Look for x = 4 on g ( x ).

          = f (0) (4, 0) is on the graph of g ( x ), so g (4) = 0.

          = −2 (0, −2) is on the graph of f ( x ), so f (0) = −2.

    g o f (−2) = g ( f (−2)) Look for x = −2 on the graph of f ( x ).

          = g (1) (−2, 1) is on the graph of f ( x ), so f (−2) = 1.

          = 4 (1, 4) is on the graph of g ( x ), so g (1) = 4.

  4.  

    Combinations of Functions and Inverse Functions Solutions

  5. One possibility is g ( x ) = x − 5 and f ( x ) = x 3 + 2.

f o g ( x ) = f ( g ( x )) = f ( x − 5) = ( x − 5) 3 + 2 = h ( x )

Find practice problems and solutions for these concepts at Combinations of Functions and Inverse Functions Practice Test.

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