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Combinations of Functions Help (page 2)

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

Example

Refer to Figure 4.1. 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 f g (3), f o g (5), and g o f (0).
• f o g (−1) = f ( g (−1)) Look for x = −1 on g ( x ).

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

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

Fig. 4.1 .

f o g (3) = f ( g (3)) Look for x = 3 on g ( x ).

= f (−2) (3, −2) is on the graph of g ( x ), so g (3) = −2.

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

f o g (5) = f ( g (5)) Look for x = 5 on g ( x ).

= f (0) (5, 0) is on the graph of g (x), so g (5) = 0.

= −1 (0, −1) is on the graph of f (x), so f (0) = −1.

g o f (0) = g ( f (0)) Look for x = 0 on f (x).

= g (−1) (0, −1) is on the graph of f ( x ), so f (0) = −1.

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

Unfortunately, finding the domain for the composition of two functions is not straightforward. The definition for the domain of f o g ( x ) is the set of all real numbers x such that g ( x ) is in the domain of f ( x ). When finding the domain f o g ( x ), begin with the domain with g ( x ). Then remove any x -value whose y -value is not in the domain for f ( x ). For example if , the y -values for g ( x ) are x + 3. We need for x + 3 to be nonzero for .

Examples

Find the domain for f o g ( x ).

The domain for g ( x ) is x ≥ 3 (from 2 x − 6 ≥ 0). Are there any x -values in [3, ∞) we cannot put into We cannot allow to be zero, so we cannot allow x = 3. The domain for f o g ( x ) is (3, ∞).

The domain for g ( x ) is x ≠ −1. Are there any x -values we need to remove from x ≠ −1? We need to find any real numbers that are not in the domain for

The denominator of this fraction is , so we cannot allow to be zero. A fraction equals zero only when the numerator is zero, so we cannot allow x − 1 to be zero. We must remove x = 1 from the domain of g(x).

The domain of f o g (x) is x ≠ −1, 1, or (−∞, −1) ∪ (−1, 1) ∪ (1, ∞). This function simplifies to , which hides the fact that we cannot let x = − 1.

Composing Three or More Functions Together

Any number of functions can be composed together. Functions can even be composed with themselves. When composing three or more functions together, we will work from the right to the left, performing one composition at a time.

Examples

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

• f ( x ) = x 3 , g ( x ) = 2 x − 5, and h ( x ) = x 2 + 1.

f o f ( x ) = f ( f ( x )) = f ( x 3 ) = ( x 3 ) 3 = x 9

For f o g o h ( x ), we will begin with g o h ( x ) = g ( h ( x )) = g ( x 2 + 1) = 2 ( x 2 + 1) − 5 = 2 x 2 − 3. Now we need to evaluate f ( x ) at 2 x 2 − 3.

f o g o h ( x ) = f ( g ( h ( x ))

= f (2 x 2 − 3) = (2 x 2 − 3) 3

• f ( x ) = 3 x + 7, g ( x ) = | x − 2|, and h ( x ) = x 4 − 5

f o f ( x ) = f ( f ( x )) = f (3 x + 7) = 3(3 x + 7) + 7 = 9 x + 28

f o g o h ( x ) = f o g ( h ( x ))

g ( h ( x )) = g ( x 4 − 5) = |( x 4 − 5) − 2| = | x 4 −7|

f o g ( h ( x )) = f ( g ( h ( x ))) = f (| x 4 − 7|)

= 3| x 4 − 7| + 7

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