72 CHAPTER 2 Functions and Their Graphs Operations on Functions Let f and g be two functions defined as ( ) ( ) = + = − f x x g x x x 1 2 and 1 Find the following functions, and determine the domain. (a) ( )( ) + f g x (b) ( )( ) − f g x (c) ( )( ) ⋅ f g x (d) ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ f g x Solution EXAMPLE 11 The domain of f is { } ≠ − x x 2 and the domain of g is { } ≠ x x 1 . (a) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) + = + = + + − = − + − + + + − = − + + + − = + − + − f g x f x g x x x x x x x x x x x x x x x x x x x x 1 2 1 1 2 1 2 2 1 1 2 2 1 3 1 2 1 2 2 The domain of + f g consists of all real numbers x that are in the domains of both f and g. The domain of + f g is { } ≠ − ≠ x x x 2, 1 . (b) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) − = − = + − − = − + − − + + − = − − − + − = − − − + − = − + + + − f g x f x g x x x x x x x x x x x x x x x x x x x x x x x x 1 2 1 1 2 1 2 2 1 1 2 2 1 1 2 1 1 2 1 2 2 2 The domain of − f g consists of all real numbers x that are in the domains of both f and g. The domain of − f g is { } ≠ − ≠ x x x 2, 1 . (c) ( )( ) ( ) ( ) ( )( ) ⋅ = ⋅ = + ⋅ − = + − f g x f x g x x x x x x x 1 2 1 2 1 The domain of ⋅ f g consists of all real numbers x that are in the domains of both f and g. The domain of ⋅ f g is { } ≠ − ≠ x x x 2, 1 . (d) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = = + − = + ⋅ − = − + f g x f x g x x x x x x x x x x 1 2 1 1 2 1 1 2 The domain of f g consists of all real numbers x for which ( ) ≠ g x 0 that are also in the domains of both f and g. That is, { } ( ) = ≠ ∩ ∩ f g x g x f g domain of 0 domain of domain of DEFINITION Quotient Function Given functions f and g, the quotient function is defined by ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = ≠ f g x f x g x g x 0
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