302 CHAPTER 2 Graphs and Functions Step 3 Find the difference quotient by dividing by h. ƒ1x + h2 - ƒ1x2 h = 4xh + 2h2 - 3h h Substitute 4xh + 2h2 - 3h for ƒ1x + h2 - ƒ1x2, from Step 2. = h14x + 2h - 32 h Factor out h. = 4x + 2h - 3 Divide out the common factor. S Now Try Exercise 55. CAUTION In Example 4, notice that the expression ƒ1x + h2 is not equivalent to ƒ1x2 + ƒ1h2. These expressions differ by 4xh. ƒ1x + h2 = 21x + h22 - 31x + h2 = 2x2 + 4xh + 2h2 - 3x - 3h ƒ1x2 + ƒ1h2 = 12x2 - 3x2 + 12h2 - 3h2 = 2x2 - 3x + 2h2 - 3h In general, for a function ƒ, ƒ1x + h2 is not equivalent to ƒ1x2 + ƒ1h2. Composition of Functions and Domain The diagram in Figure 94 shows a function g that assigns to each x in its domain a value g1x2. Then another function ƒ assigns to each g1x2 in its domain a value ƒ1g1x22. This two-step process takes an element x and produces a corresponding element ƒ1g1x22. Input x Output f(g(x)) g g(x) f Function Function Figure 94 The function with y-values ƒ1g1x22 is the composition of functions ƒ and g, which is written ƒ° g and read “ƒ of g” or “ƒ compose g.” Composition of Functions and Domain If ƒ and g are functions, then the composite function, or composition, of ƒ and g is defined by 1 ƒ° g2 1x2 =ƒ1g1x2 2. The domain of ƒ° g is the set of all numbers x in the domain of g such that g1x2 is in the domain of ƒ. As a real-life example of how composite functions occur, consider the following retail situation. A $40 pair of blue jeans is on sale for 25% off. If we purchase the jeans before noon, they are an additional 10% off. What is the final sale price of the jeans? We might be tempted to say that the jeans are 35% off and calculate $4010.352 = $14, giving a final sale price of $40 - $14 = $26. Incorrect
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