Function f Function f-1 Ch. 9.4 Inverse Functions Page 1 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 Function f Output f(x) Input f(x) Function f-1 Output x

if x1 is not equal to x2 (x1 and x2 any elements of the domain) Page 2 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 x1 is not equal to x2 (x1 and x2 any elements of the domain) then f(x1) is not equal to f(x2). In other words, for any two ordered pairs (x1,y1) and (x2, y2) , where y1 = f(x1) and y2 = f(x2), Then if x1 ≠ x2, then y1 ≠ y2. Similarly, if f(x1) = f(x2), then it must be that x1 = x2. 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=x2-4 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=x2-4 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 = x2 A function is 1-to-1 over a certain interval only if it is constantly decreasing or constantly increasing over that interval. y=x2-4 is 1-to-1 over the intervals (-∞,0) and (0, ∞) Which one do we choose? We need to have a specific value

f(x) = 2x + 3 y = x f-1(x) = ½ (x-3) Page 3 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 – 3 + 3 = x f(x) = 2x + 3 (1,5) y = x f-1(x) = ½ (x-3) (5,1)

Step 1: Interchange x and y Page 4 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

Example Find the domain and range of 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}