1. Inverse Functions By Dr. Carol A. Marinas

2. A function is a relation when each x-value is paired with only 1 y-value. (Vertical Line Test) A function f is said to be one-to-one when each y-value is paired with only 1 x-value. (Horizontal Line Test) Which of the following are one-to-one functions? {(1, 1), (2, 4), (3, 9), (4, 16)} {( – 2, 4), ( – 1, 1), (0, 0), (1, 1)} one-to-one not one-to-one

3. Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one. Use the graph to determine whether the following function is one-to-one. x y (0,0)

4. Use the graph to determine whether the following function is one-to-one. x y (1,1) (–1, –1)

5. Theorem: A function that is increasing on an interval I is one-to-one on I. Theorem: A function that is decreasing on an interval I is one-to-one on I. Example: The function in increasing on the interval to the right of zero. Therefore, it is one-to- one on that interval. Verify this statement with the horizontal line test.

6. Let f denote a one-to-one function y = f (x). The inverse of f, denoted f -1, is a function such that f -1 (f (x)) = x for every x in the domain f and f (f -1 (x)) = x for every x in the domain of f -1. In other words, the function f maps each x in its domain to a unique y in its range. The inverse function f -1 maps each y in the range back to the x in the domain. See Examples: p. 400 (Sullivan Text)

7. Domain of fRange of f

8. The graph of a function f and the graph of its inverse are symmetric with respect to the line y=x. y = x f f -1

9. Finding the inverse of a function. Since the domain of f is the range of f -1 and visa versa, we find the inverse of f by interchanging x and y. Example: Find the inverse of fx x x(),    5 3 3

10. To determine if two given functions are inverses, we use the definition of inverse functions. If two functions are inverses, then f -1 (f (x)) = x and f (f -1 (x)) = x.