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Combinations of Functions: Composite Functions Section 2.6 College Algebra, MATH 171 Mr. Keltner “Combining” Functions If we have two functions, call them f(x) and g(x), then we can combine them to form new functions: (f + g)(x) (f - g)(x) (f g)(x) (f/g)(x) This is done very similar to how we combine real numbers, or combine like terms (terms with the same variables and exponents, such as -5x2 and 8x2) in a long expression. Combining Functions: What it looks like We can define the function (f + g)(x) as simple as combining the expressions for f(x) and g(x), or like this: (f + g)(x) = f(x) + g(x) The new function, f + g, is called the sum of the functions f and g. The domain of our new function is all the x-values that were in the domain of both f(x) and g(x). Combining Functions: A little name-calling Similar to defining the function (f + g)(x) as the sum of the functions f(x) and g(x), we can also define these other combinations of functions f(x) and g(x): (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f g)(x) = f(x) g(x) (f/g)(x) = f(x) ÷ g(x), where g(x) ≠ 0 With the quotient, you must also remember that we do not want to divide by zero, and determine the values of the domain accordingly. Example 1 Assume f x 5x 3 and gx 11x Find the following: 1 f(x) + g(x) f(x) - g(x) Evaluate each when x = 2. 1 3 Example 2 Let f x 8x and gx 2x Find the following: 5 6 f(x) g(x) f x g x Find the domain of each new function. Composition of Functions When we show that one function’s result depends on another function’s result, we can describe this by working with a composition of functions. If we say that h(x) is a composition of functions f(x) and g(x), we can write this as: hx f gx It is also sometimes written as (f g)(x). Composition of Functions: Importance of Order Just because f(g(x)) looks like multiplication does not mean we can apply properties of multiplication when working with them. In other words, there are only certain special instances where f(g(x)) = g(f(x)). This is discussed in section 2.7 in particular and points out a unique feature of the functions f(x) and g(x). So, in general, we conclude that f(g(x)) ≠ g(f(x)). The next example illustrates why this is true. Composition of Functions “Practical” Example Kohl’s sends us a BUNCH of coupons as a “Most Valuable Customer” promotion. (I think we’re the most awesomest, but I don’t teach English, obviously) We receive two coupons: If there is no fine print saying we can’t use BOTH coupons, in which order should we ask the clerk to ring them up? $10 off any purchase 20% off any one day’s purchases $10 off first, or 20% off first? Write each scenario as an equation to see what you still pay for. Try a $45 purchase for example. Steps in Composition of Functions Example 3: Let f(x) = -3x - 4 and g(x) = x2 - 1. Find the value of f(g(-3)). Evaluate g(-3) by inserting -3 into g(x) and simplifying to ______. Now, use that value in f(x) to finish up. This means we are evaluating f(_____). 1) 2) 3) Evaluate (or substitute in) the function in the inner-most parentheses first. Using function notation, substitute this value (or expression) wherever you see a variable in the function outside the parentheses. Simplify, when applicable. Example 4 Let f(x) = 6x-2 and g(x) = 4x + 5. Find each of the following: f(g(x)) g(f(x)) g(g(x)) Example 5: Composition of 3 functions Find f g h given the following information about f(x), g(x), and h(x): x f x x 1 gx x10 h x x 3 Recognizing a Composition of Functions Given the function F x x 2 2x 3 , find functions f(x) and g(x) such that F(x) = f g. For a composition of functions, remember f g = f(g(x)). So, if we can figure out which quantity we need to calculate first, we also know our inside function, g(x). Assessment Pgs. 238-240: #’s 3 - 36, 45 - 63, multiples of 3