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2.7 Function Operations and Composition Operations on Functions • Given two functions f and g, then for all values of x for which both f(x) and g(x) are defined, the functions f + g, f - g, fg, and are defined as follows: – (f + g)(x) = f(x) + g(x) Sum – (f - g)(x) = f(x) - g(x) Difference – (fg)(x) = f(x) × g(x) Product – Quotient Example Let f(x) = x + 1 and g(x) = 2x2 - 3. • Find (f + g)(3) • Find (fg)(-2) Example continued Solution: • Since f(3) = 4 and g(3) = 15, (f + g)(3) = f(3) + g(3) (f + g)(x) = f(x) + g(x) = 4 + 15 = 19 • Since f(-2) = -1 and g(-2) = 5, (fg)(-2) = f(-2) × g(-2) (fg)(x) = f(x) × g(x) = -1 × 5 = -5 Domains For functions f and g, the domain of f + g, f - g, and fg include all real numbers in the intersection of the domains of f and g, while the domain of includes those real numbers in the intersection of the domains of f and g for which g(x) ¹ 0. Example: Let f(x) = 3x - 5 and g(x) = • Find (f - g)(x) • Find • Give the domains of each function. Domains continued Solutions: • (f - g)x = f(x) – g(x) = • • In part (a), the domain of f is the set of all real numbers (- , ), and the domain of g, since includes just the real numbers to make 3x - 2 nonnegative. That is , so . Domains continued The domain of g is . The domain of f - g is the intersection of the domains of f and g, which is . The domain of includes those real numbers in the intersection above for which ; that is, the domain of is . The Difference Quotient Suppose the point P lies on the graph of y = f(x), and h is a positive number. If we let (x, f(x)) denote the coordinates of P and (x + h, f(x + h)) denote the coordinates of Q, then the line joining P and Q has slope This expression is called the difference quotient. Example Let f(x) = x2 + 4x. Find the difference quotient and simplify the expression. Solution: Step 1 Find f(x + h) Replace x in f(x) with x + h. f(x + h) = (x + h)2 + 4(x + h) Step 2 Find f(x + h) - f(x) f(x + h) - f(x) = [ (x + h)2 + 4(x + h)] - (x2 + 4x) Substitute. = x2 + 2xh + h2 + 4(x + h) - (x2 + 4x) Square x + h. = x2 + 2xh + h2 + 4x + 4h - x2 - 4x = 2xh + h2 + 4h Example continued Step 3 Find the difference quotient. Composition of Functions If f and g are functions, then the composite functions, or composition, of g and f is defined by The domain of is the set of all numbers x in the domain of f such that f(x) is in the domain of g. Example Let f(x) = 3x - 1 and g(x) = x + 5. Find each composition. • Solution: First find g(2). Since g(x) = x + 5, g(2) = 2 + 5 = 7. Now find = f[g(2)] = f(7): f[g(2)] = f(7) = 3(7) - 1 = 20. Example continued • Solution: First find f(2). Since f(x) = 3x - 1 f(2) = (3(2) - 1) = 5 Now find = g[f(2)] = g(5) g[f(2)] = g(5) = 5 + 5 = 10. Finding Composite Functions • Example: Let f(x) = x2 and g(x) = x - 2. Find and . • Solution: Composite Functions • The previous example shows it is not always true that . – In fact, the composite functions are equal only for a special class of functions. Finding Functions That Form a Given Composite Example: Find functions f and g such that Solution: Note the repeated quantity x2 + 3. If we choose g(x) = x2 + 3 and f(x) = x3 - 2x + 7 then