SECTION 5.1 Composite Functions 275 In determining the domain of the composite function f g x f g x , ( ) ( ) ( ) ( ) = keep the following two thoughts in mind about the input x. • Any x not in the domain of g must be excluded. • Any x for which g x( ) is not in the domain of f must be excluded. (b) ( ) ( ) ( ) ( ) ( ) ( ) = = + − = + − + = + − + = + + g f x g f x g x x x x x x x x 3 1 2 3 1 3 2 6 2 3 2 6 1 2 2 2 2 Because the domains of both f and g are the set of all real numbers, the domain of g f is the set of all real numbers. ( ) ( ) = + − = + f x x x g x x 3 1 2 3 2 ↑ ↑ Example 2 illustrates that, in general, f g g f. ≠ Sometimes f g does equal g f, as we shall see in Example 5. Look back at Figure 2 on page 273. Finding the Domain of f g Find the domain of f g if f x x 1 2 ( ) = + and g x x 4 1 . ( ) = − Solution EXAMPLE 3 For f g x f g x , ( ) ( ) ( ) ( ) = first note that the domain of g is x x| 1 , { } ≠ so 1 is excluded from the domain of f g. Next note that the domain of f is x x| 2 , { } ≠ − which means that g x( ) cannot equal 2. − Solve the equation g x 2 ( ) = − to determine what additional value(s) of x to exclude. ( ) − =− =− − =− + =− =− x x x x x 4 1 2 4 2 1 4 2 2 2 2 1 Also exclude 1− from the domain of f g. The domain of f g is x x x | 1, 1 . { } ≠ − ≠ Check: For x g x x 1, 4 1 ( ) = = − is not defined, so f g f g 1 1 ( ) ( ) ( ) ( ) = is not defined. For x g 1, 1 2, ( ) = − − = − and f g f g f 1 1 2 ( ) ( ) ( ) ( ) ( ) − = − = − is not defined. ( ) =− g x 2 Multiply both sides by −x 1. Finding a Composite Function and Its Domain Suppose f x x 1 2 ( ) = + and g x x 4 1 . ( ) = − Find: (a) f g (b) f f Then find the domain of each composite function. (continued) EXAMPLE 4 Solution The domain of f is x x| 2 { } ≠ − and the domain of g is x x| 1 . { } ≠ (a) f g x f g x f x x x x x x x x 4 1 1 4 1 2 1 4 2 1 1 2 2 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = − = − + = − + − = − + = − + g x x f x x x x 4 1 1 2 Multiplyby 1 1 . ( ) ( ) = − = + − − ↑ ↑ ↑
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