303 2.8 Function Operations and Composition $26 is not correct. To find the final sale price, we must first find the price after taking 25% off and then take an additional 10% off that price. See Figure 95. $40 Input original price $27 Output sale price g(40) = 40 – 0.25(40) = 40 – 10 = 30 $30 f(30) = 30 – 0.10(30) = 30 – 3 = 27 Function g takes 25% off. Function f takes an additional 10% off. g(x) = x – 0.25x ( f ° g)(40) = f( g(40)) = f(30) = 27 f(x) = x – 0.10x Figure 95 EXAMPLE 5 Evaluating Composite Functions Let ƒ1x2 = 2x - 1 and g1x2 = 4 x - 1 . (a) Find 1ƒ∘ g2122. (b) Find 1g∘ ƒ21-32. SOLUTION (a) First find g122: g122 = 4 2 - 1 = 4 1 = 4. Now find 1ƒ∘ g2122. 1ƒ∘ g2122 = ƒ1g1222 Definition of composition = ƒ142 g122 = 4 from above = 2142 - 1 Substitute 4 for x in ƒ1x2 = 2x - 1. = 7 Simplify. (b) 1g∘ ƒ21-32 = g1ƒ1-322 Definition of composition = g321-32 - 14 ƒ1-32 = 21-32 - 1 = g1-72 Multiply, and then subtract. = 4 -7 - 1 Substitute -7 for x in g1x2 = 4 x - 1 . = - 1 2 Subtract in the denominator. Write in lowest terms. S Now Try Exercise 57. EXAMPLE 6 Determining Composite Functions andTheir Domains Given that ƒ1x2 = 2x and g1x2 = 4x + 2, find each of the following. (a) 1ƒ∘ g21x2 and its domain (b) 1g∘ ƒ21x2 and its domain SOLUTION (a) 1ƒ∘ g21x2 = ƒ1g1x22 Definition of composition = ƒ14x + 22 g1x2 = 4x + 2 = 24x + 2 ƒ1x2 = 2x The screens show how a graphing calculator evaluates the expressions in Example 5.
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