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Section 4.2 Operations with Functions Objectives: 1. To add, subtract, multiply, and divide functions. 2. To find the composition of functions. EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/ (x). g (f +g)(x) = f(x) + g(x) = (x2 – 9) + (x + 3) = x2 + x – 6 EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/ (x). g (f – g)(x) = f(x) – g(x) = (x2 – 9) – (x + 3) = x2 – 9 – x – 3 = x2 – x – 12 EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/ (x). g (fg)(x) = f(x)g(x) = (x2 – 9)(x + 3) = x3 + 3x2 – 9x – 27 EXAMPLE 1 Let f(x) = x2 – 9 and g(x) = x + 3. Find (f + g)(x), (f – g)(x), fg(x), and f/ (x). g 2–9 x f/ (x) = g x+3 (x – 3)(x + 3) = x+3 = x – 3, if x ≠ -3 EXAMPLE 2 Let f(x) = 5x – 7 and g(x) = x2 + 3x – 2. Find f(a + b), f(x2 – 9), g(4a), and g(3x + 1) f(a + b) = 5(a + b) – 7 = 5a + 5b – 7 f(x2 – 9) = 5(x2 – 9) – 7 = 5x2 – 45 – 7 = 5x2 – 52 EXAMPLE 2 Let f(x) = 5x – 7 and g(x) = x2 + 3x – 2. Find f(a + b), f(x2 – 9), g(4a), and g(3x + 1) g(4a) = (4a)2 + 3(4a) – 2 = 16a2 + 12a – 2 g(3x + 1) = (3x + 1)2 + 3(3x + 1) – 2 = 9x2 + 6x + 1 + 9x + 3 – 2 = 9x2 + 15x + 2 Definition Composition An operation that substitutes the second function into the first function. In symbols: g ◦ f = g(f(x)). Read g ◦ f as “the composition of g with f” or “g composed with f”. Mapping diagrams provide a useful representation of composition. Let f(x) = 3x – 5 and g(x) = x2 – 9, and let Df = {5, 3, -1, 0}. g◦f -1 0 3 5 Df f 3x – 5 -8 -5 4 10 Rf Dg g x2 – 9 55 16 7 91 Rg From the circle diagram you can see that g ◦ f = {(-1, 55), (0, 16), (3, 7), (5, 91)}. A function rule for the composition of two functions could also be used to find the ordered pairs. The rule can be found from the rules of the original functions. To find the rule for the composite function substitute the second function into the first as illustrated in Example 3. Use the rule to check that it obtains the same set of ordered pairs: {(-1, 55), (0, 16), (3, 7), (5, 91)}. Check for the ordered pair (3, 7). (g ◦ f)(x) = 9x2 – 30x + 16 (g ◦ f)(3) = 9(32) – 30(3) + 16 = 81 – 90 + 16 =7 EXAMPLE 3 Find (g ◦ f)(x) if f(x) = 3x – 5 and g(x) = x2 – 9. (g ◦ f)(x) = g(f(x)) = g(3x – 5) = (3x – 5)2 – 9 = 9x2 – 30x + 25 – 9 = 9x2 – 30x + 16 Homework: pp. 181-182 ►A. Exercises Let f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following. 3. f(x2) ►A. Exercises Let f(x) = -2x + 7, g(x) = 5x2, h(x) = x – 9. Evaluate the following. 5. g(3a + b) ►A. Exercises If f(x) = -2x + 7, g(x) = 5x2, and h(x) = x – 9, perform the following operations. 11. fh(x) ►B. Exercises Let f(x) = x, g(x) = x – 7, h(x) = x2 + 8, k(x) = 5x – 4. Find the function rules for the composition functions. 19. g ◦ h ►B. Exercises Let f(x) = x, g(x) = x – 7, h(x) = x2 + 8, k(x) = 5x – 4. Find the function rules for the composition functions. 23. k ◦ f ■ Cumulative Review 36. Find the amount in a savings account after five years if $2000 is invested at 5% interest compounded quarterly. ■ Cumulative Review 37. Use the exponential growth function f(x) = C ● 2x to find the number of bacteria in a culture after 8 days if there were originally 20 bacteria. ■ Cumulative Review 38. Graph the piece function -1 if x -1 f(x) = x3 if -1 x 1 ½x if x 1 ■ Cumulative Review 39. Find the slope of a line perpendicular to 3x + 5y = 6. ■ Cumulative Review 40. Find A for right triangle ABC with C = 90°, a = 2, and b = 3.