1.6 Inverse Functions The graph of an inverse is a reflection of the original function across the line y x , which means if the point a, b is on the original then the point b, a is on the inverse. Remember that a function is a mapping of input values to output values. An inverse of a function maps outputs back to inputs. This means that the domain of the inverse is the range of the original, and the range of the inverse is the domain of the original. Exploration: Graph f ( x) x 2 below, then graph its inverse by plotting points. Is the result a function? If the original function is one-to-one (meaning no two x-values map to the same y-value), its graph passes the horizontal line test and it is possible to find an inverse function. Since not all functions are one-to-one that means that some functions don’t have inverse functions. Note: The expression f 1 means “inverse of f”. In this notation, it does not mean that -1 is a power, and therefore it doesn’t mean f 1 is the reciprocal of f. You cannot find the inverse of a function by just pressing the x 1 button on your calculator! Examples: 1. Given f (2) 4, f (1) 3, f (0) 2 and f (2) 3 . What is f 1 (2) ? 2. Consider f ( x) x 2 3x 2 . Determine if it is one-to-one. Functions that are inverses of each other have the effect of “undoing” each other. That means that if you do the composition of the two functions (in either order), you will obtain the identity function. Do you know what that means? Example: Verify that f ( x) 5 x 8 and g ( x ) x 8 are inverses of each other. 5 Finding an inverse function: 1. Use the horizontal line test to decide whether f ( x) has an inverse function. 2. Interchange x and y and then solve for y. 3. Replace y with f 1 x in the new equation. 4. State the domain of f 1 x . Note: This is equivalent the range of f ( x) . Examples: Find the inverse of the following functions. 1. f ( x) 3 x 6 2. g ( x) x7 7 3. f ( x) x 4 ; x 0 4. f ( x) x 6 Why is the domain restricted? Could you restrict the domain any other way?
