304 CHAPTER 2 Graphs and Functions The domain and range of g are both the set of all real numbers, 1-∞, ∞2. The domain of ƒ is the set of all nonnegative real numbers, 30, ∞2. Thus, g1x2, which is defined as 4x + 2, must be greater than or equal to 0. 4x + 2 Ú 0 Solve the inequality. x Ú - 1 2 Subtract 2. Divide by 4. Therefore, the domain of ƒ∘ g is C - 1 2 , ∞B. The radicand must be nonnegative. (b) 1g∘ ƒ21x2 = g1ƒ1x22 Definition of composition = g A 2x B ƒ1x2 = 2x = 42x + 2 g1x2 = 4x + 2 The domain and range of ƒ are both the set of all nonnegative real numbers, 30, ∞2. The domain of g is the set of all real numbers, 1-∞, ∞2. Therefore, the domain of g∘ ƒ is 30, ∞2. S Now Try Exercise 75. EXAMPLE 7 Determining Composite Functions andTheir Domains Given that ƒ1x2 = 6 x - 3 and g1x2 = 1 x , find each of the following. (a) 1ƒ∘ g21x2 and its domain (b) 1g∘ ƒ21x2 and its domain SOLUTION (a) 1ƒ∘ g21x2 = ƒ1g1x22 By definition = ƒ a 1 xb g1x2 = 1 x = 6 1 x - 3 ƒ1x2 = 6 x - 3 = 6x 1 - 3x Multiply the numerator and denominator by x. The domain of g is the set of all real numbers except 0, which makes g1x2 undefined. The domain of ƒ is the set of all real numbers except 3. The expression for g1x2, therefore, cannot equal 3. We determine the value that makes g1x2 = 3 and exclude it from the domain of ƒ∘ g. 1 x = 3 The solution must be excluded. 1 = 3x Multiply by x. x = 1 3 Divide by 3. Therefore, the domain of ƒ∘ g is the set of all real numbers except 0 and 1 3 . 1-∞, 02 ´a0, 1 3b ´a 1 3 , ∞b.
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