Sullivan PreCalculus Section 4.2 Inverse Functions Objectives of this Section Determine the Inverse of a Function Obtain the Graph of the Inverse Function From the Graph of the Function Find the Inverse Function f -1
one-to-one not one-to-one A function f is said to be one-to-one if, for any choice of numbers x1 and x2, x1 x2, in the domain of f, then f (x1) f (x2). Which of the following are one - to - one functions? {(1, 1), (2, 4), (3, 9), (4, 16)} one-to-one {(-2, 4), (-1, 1), (0, 0), (1, 1)} not one-to-one
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)
Use the graph to determine whether the following function is one - to - one. x y (1,1) (-1,-1)
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.
Let f denote a one-to-one function y = f (x) 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.
Domain of f Range of f
The graph of a function f and the graph of its inverse are symmetric with respect to the line y=x.
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. f x ( ) , = - ¹ 5 3 Example: Find the inverse of
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.