MAC1140, MAC11476.2: One-to-One Functions; Inverse FunctionsIf f(x) = y is a function, its inverse f –1(x) is f(y) = x, where y becomes the domain and x becomes the range. If for each y in the domain of the inverse function there is a unique x in the range, it is a one-to-one function. If ahorizontal line intersects the graph of a function f no more than once, then f is one-to-one. Only one-to-one functions have inverses. We can verify that f and f –1 are inverses showing that f ( f –1 (x)) = f -1 ( f (x)) = x. The graph of a function f and its inverse f –1 are symmetric with respect to the line y = x. To find the inverse of a function y = f(x), interchange x and y to obtain x = f(y) and solve for y in terms of x.Find the inverse and determine whether the inverse is a function:1. [(-2, 5), (-1, 3), (3, 7), (4, 12)2. y = x2Use the horizontal line test to see whether f is one-to-one; if yes, graph its inverse:(3) y = x - 3 (4) (5) y = f(x) = -2x + 3 Verify that the functions f and g are inverses of each other:6. f(x) = 3 – 2x; g(x) = 123( )x7. f(x) = 3 5x , 253xg xThe following functions are one-to-one. Find the inverse and state the domain and range of each.8.f xxx( ) 239. 24, 0f x x x #9f(x) f -1(x)#9f(x) f -1(x)Domain DomainRange RangeMAC1140, MAC1147TRY THESE (6.2) Which of the following graphs are one-to-one? Draw the graph of the inverse function f-1 for those that are one-to-one.1. a. b. c. d.Verify that the functions f and g are inverses of each other.2. 5 3 5;2 3 1 2x xf x g xx x Function f is one-to-one. Find its inverse; check your answer. State the domain of f and find its range using f-1.3.( )+= >-23, 03 4xf x xx#3f(x) f
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