306 CHAPTER 2 Graphs and Functions In calculus it is sometimes necessary to treat a function as a composition of two functions. The next example shows how this can be done. EXAMPLE 9 Finding FunctionsThat Form a Given Composite Find functions ƒ and g such that 1ƒ∘ g21x2 = 1x2 - 523 - 41x2 - 52 + 3. SOLUTION Note the repeated quantity x2 - 5. If we choose g1x2 = x2 - 5 and ƒ1x2 = x3 - 4x + 3, then we have the following. 1ƒ∘ g21x2 = ƒ1g1x22 By definition = ƒ1x2 - 52 g1x2 = x2 - 5 = 1x2 - 523 - 41x2 - 52 + 3 Use the rule for ƒ. There are other pairs of functions ƒ and g that also satisfy these conditions. For instance, ƒ1x2 = 1x - 523 - 41x - 52 + 3 and g1x2 = x2. S Now Try Exercise 99. 2.8 Exercises CONCEPT PREVIEW Without using paper and pencil, evaluate each expression given the following functions. ƒ1x2 = x + 1 and g1x2 = x2 1. 1ƒ + g2122 2. 1ƒ - g2122 3. 1ƒg2122 4. a ƒ gb 122 5. 1ƒ∘ g2122 6. 1g∘ ƒ2122 CONCEPT PREVIEW Refer to functions ƒ and g as described in Exercises 1–6, and find the following. 7. domain of ƒ 8. domain of g 9. domain of ƒ + g 10. domain of ƒ g For the pair of functions defined, find 1ƒ + g21x2, 1ƒ - g21x2, 1ƒg21x2, and A f gB1x2. Give the domain of each. See Example 2. 19. ƒ1x2 = 3x + 4, g1x2 = 2x - 5 20. ƒ1x2 =6 - 3x, g1x2 = -4x + 1 21. ƒ1x2 = 2x2 - 3x, g1x2 = x2 - x + 3 22. ƒ1x2 =4x2 +2x, g1x2 = x2 - 3x + 2 23. ƒ1x2 = 24x - 1, g1x2 = 1 x 24. ƒ1x2 = 25x - 4, g1x2 = - 1 x Let ƒ1x2 = x2 + 3 and g1x2 = -2x + 6. Find each of the following. See Example 1. 11. 1ƒ + g2132 12. 1ƒ + g21-52 13. 1ƒ - g21-12 14. 1ƒ - g2142 15. 1ƒg2142 16. 1ƒg21-32 17. a ƒ gb1-12 18. a ƒ gb152
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