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Lesson 102: Sums of Functions, Products of Functions Recall that functional notation has several advantages. The first is that it allows us to identify the equations being considered. For instance, if we have the three equations y=x+3 2 y=x–6 2 y = 2x + 5 We can name them as equation f, equation g, and equation h. if we use functional notation, we would use f(x), g(x), and h(x) instead of y. f(x) = x + 3 2 g(x) = x – 6 2 h(x) = 2x + 5 If we add two of these equations, we get an equation for the sum. The sum of equations f and h is f(x) = x + 3 h(x) = 2x + 5 2 f(x) + h(x) = 2x + x + 8 2 We see that f(x) + h(x) means that we have added the f equation and the h equation. Often we use the following notation to mean the same thing. (f + h)(x) This means that we have added the f equation to the h equation. The domain for the new equation is all numbers that were common to both of the original domains. Example: Given f(x) = x + 3; D = {Reals}, 2 and g(x) = x – 6; D = {Integers}, find (f + g)(2). Answer: f(2) = 5 g(2) = -2 (f + g)(2) = 3 Example: Given f(x) = x + 3; D = {Reals}, 2 and h(x) = 2x + 5; D = {Negative Integers}, find (f + h)(5). Answer: Since 5 is not a member of the domain of h(x) we say that the problem has no answer or an empty set. When we multiply two functions, the product is also a function. If we have the equations h(x) = x + 3; D = {Reals} 2 g(x) = x – 6; D = {negative integers} And we multiply the h equation by the g equation, we get the product h(x)g(x). h(x)g(x) = (x + 3)(x –2 6) h(x)g(x) = x + 3x – 6x 18 3 2 Instead of writing h(x)g(x) to designate the product of the two functions, we find it convenient to write hg(x) The notation hg means that the h equation has been multiplied by the g equation in the same way that (h + g)(x) Means that the h equation and the g equation have been added. Example: Find hg(-4) where h(x) = x + 3; D = 2 {reals}, and g(x) = x – 6; D = {Negative integers} Answer: hg(-4) = -10 Example: Find fg(-4) where f(x) = x + 3; D = {Reals}, and g(x) = x – 5; D = {Positive integers}. Answer: -4 is not a member of the domain of g(x); so it is not a member of the domain fg(x). Empty set. HW: Lesson 102 #1-30