Inverses Algebraically

2 Objectives I can find the inverse of a relation algebraically

3 NOTATION FOR THE INVERSE FUNCTION We use the notation for the inverse of f(x). NOTE:does NOT mean

4 Inverses The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. Example: { (1, 2), (3, -1), (5, 4)} is a relation { (2, 1), (-1, 3), (4, 5) is the inverse.

5 The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function. Ordered Pairs

6 DomainRange Inverse relation x = |y| x y Domain Range x y Function y = |x| + 1 Every function y = f (x) has an inverse relation x = f (y). The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}. x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}. The inverse relation is not a function. It pairs 2 to both -1 and +1. Inverse Relation

7 FINDING A FORMULA FOR AN INVERSE FUNCTION To find a formula for the inverse given an equation for a one-to-one function: 1. Replace f (x) with y. 2. Interchange x and y. 3. Solve the resulting equation for y. 4. Replace y with f -1 (x) if the inverse is a function.

8 Example: Find the inverse relation algebraically for the function f (x) = 3x + 2. y = 3x + 2 Original equation defining f x = 3y + 2 Switch x and y. 3y + 2 = x Reverse sides of the equation. y = Solve for y. To find the inverse of a relation algebraically, interchange x and y and solve for y. Example: Inverse Relation Algebraically

9 Example 2: Find the inverse relation algebraically for the function f (x) = x y = x Original equation defining f x = y Switch x and y. y 2 = x - 6 Reverse sides and subtract 6 y = Solve for y. Example: Inverse Relation Algebraically

Find the inverse function “algebraically” f(x) = 6 Exchange the x & y values x = 6 Is the inverse a function? NO, because it is a vertical line. You can find the inverse but not the inverse function.

11 TESTING FOR A ONE-TO-ONE FUNCTION Horizontal Line Test: A function is one-to- one (and has an inverse function) if and only if no horizontal line touches its graph more than once.

12 x y 2 2 Horizontal Line Test A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point. y = 7 Example: The function y = x 2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7). (0, 7) (4, 7) Horizontal Line Test

13 one-to-one Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x 3 b) y = x 3 + 3x 2 – x – 1 not one-to-one x y x y Example: Horizontal Line Test

14 y = f(x) y = x y = f -1 (x) Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. The graph of f passes the horizontal line test. The inverse relation is a function. Reflect the graph of f in the line y = x to produce the graph of f -1. x y Example: Determine Inverse Function

Homework WS