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Ch. 9.4 Inverse Functions What is an inverse function? Do all functions have inverses? An inverse function, f -1, is a kind of “undoing” function. If the initial function, f, takes the element a to the element b, then the inverse function takes the element b back to the element a. The domain of f = the range of f -1. The range of f = the domain of f -1. The graph of f and the graph of f -1 are symmetric with respect to the line y=x. (For example, if (a,b) is on the graph of f(x), then (b,a) is on the graph of f -1 (x). The composite of one function with its inverse becomes the identity function. That is, if you input x in the f function “machine”, you output f(x). If you input f(x) in the f -1 function “machine”, you output x. f -1 (f(x)) = x and also f - (f -1 (x)) = x Input x Output f(x) Function f Input f(x) Output x Function f -1 Page 1

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Only one-to-one functions have inverses. Not all functions have inverses because all functions are not one-to-one functions. Definition of a one-to-one function: A function is a one-to-one if no two different elements in the domain have the same element in the range. The definition of a one-to-one function can be written algebraically as follows: A function f(x) is one-to-one if x 1 is not equal to x 2 (x 1 and x 2 any elements of the domain) then f(x 1 ) is not equal to f(x 2 ). In other words, for any two ordered pairs (x 1,y 1 ) and (x 2, y 2 ), where y 1 = f(x 1 ) and y 2 = f(x 2 ), Then if x 1 ≠ x 2, then y 1 ≠ y 2. Similarly, if f(x 1 ) = f(x 2 ), then it must be that x 1 = x 2. Just as we had a vertical line test to test if a graph represents a function, there is a horizontal line test to test if a function is 1-to-1. Horizontal Line Test Theorem If every horizontal line intersects the graph of a function f in at most one point, then f is 1-to-1. Below is the graph of y=x 2 -4 For y=12, there are two possible x’s. x=-4, and x=4. (-4,12) (4,12) However, for each x there is only one possible y, so y=x is a function. Does not pass Horizontal Line Test Therefore, this function is not 1-to-1. What would the inverse fuction of y = x2 -4 be? Solve for x. y + 4 = x 2 Which one do we choose? We need to have a specific value A function is 1-to-1 over a certain interval only if it is constantly decreasing or constantly increasing over that interval. y=x 2 -4 is 1-to-1 over the intervals (-∞,0) and (0, ∞) Page 2

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Example Finding the inverse function Find the inverse of f(x) = 2x + 3. Step 1: let y = f(x) then interchange the variables x and y. y = 2x + 3 Interchange x and y. x = 2y + 3 Step 2: Solve for y in this new equation. x = 2y + 3 and set y = f -1 (x) 2y = x – 3 y = ½ (x – 3) f -1 (x) = ½ (x – 3) Step 3: Check the result by showing that f -1 (f(x)) = x and also f - (f -1 (x)) = x Plug in f(x) = 2x + 3 as the input in f -1 (x) = ½ (x – 3) f -1 (f(x)) = f -1 (2x+3) = ½ ((2x+3) – 3) = ½ (2x + 3 – 3) = ½ (2x) = x Now plug in f -1 (x) = ½ (x – 3) as the input for f(x) = 2x + 3 f - (f -1 (x)) = f(½ (x-3)) = 2(½ (x-3)) + 3 = 1(x-3) + 3 = x – = x f(x) = 2x + 3 y = x f -1 (x) = ½ (x-3) (1,5) (5,1) Page 3

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Example The following function is one-to-one. Find its inverse and check the result. Step 1: Interchange x and y Step 2: Solve for y by cross-multiplying Step 3: Check if f -1 (f(x)) = x Input for f -1 Page 4

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Page 5 Example Find the domain and range of The domain is easy to find. We know f(x) would be undefined if x=1, so domain includes all real numbers except x=1. {x|x≠1} Are there any limitations on the range? To find range, find the domain of the inverse of f. f -1 (x) is undefined when x=2, so range of f is all real numbers except x=2. {x|x≠2}

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