SECTION 5.2 One-to-One Functions; Inverse Functions 285 Figure 12 illustrates the theorem. Once the graph of f is known, the graph of −f 1 can be obtained by reflecting the graph of f about the line = y x. 3 Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function Suppose ( ) a b , is a point on the graph of a one-to-one function f defined by ( ) = y f x . Then ( ) = b f a . This means that ( ) = − a f b , 1 so ( ) b a , is a point on the graph of the inverse function −f .1 The relationship between the point ( ) a b , on f and the point ( ) b a , on −f 1 is shown in Figure 11. The line segment with endpoints ( ) a b , and ( ) b a , is perpendicular to the line = y x and is bisected by the line = y x. (Do you see why?) It follows that the point ( ) b a , on −f 1 is the reflection about the line = y x of the point ( ) a b , on f. Figure 11 x y 5 x y b a b a (b, a) (a, b) THEOREM The graph of a one-to-one function f and the graph of its inverse function −f 1 are symmetric with respect to the line = y x. Figure 12 y 5 x y 5 f(x) y 5 f 21(x) (a3, b3) (a2, b2) (a1, b1) (b3, a3) (b2, a2) (b1, a1) x y Now Work PROBLEM 35 Graphing an Inverse Function The graph in Figure 13(a) shows a one-to-one function ( ) = y f x . Draw the graph of its inverse. EXAMPLE 5 Solution First add the graph of = y x to Figure 13(a). Since the points ( ) ( ) − − − 2, 1 , 1, 0 ,and ( ) 2, 1 are on the graph of f, the points ( ) ( ) − − − 1, 2 , 0, 1 ,and ( ) 1, 2 must be on the graph of −f .1 Keeping in mind that the graph of −f 1 is the reflection about the line = y x of the graph of f, graph −f .1 See Figure 13(b). Figure 13 x y 23 3 (2, 1) (21, 0) (22, 21) 3 23 y 5 f(x) (a) x y 23 3 (2, 1) (1, 2) (21, 0) (22, 21) (0, 21) (21, 22) 3 23 y 5 f(x) y 5 f 21(x) (b) y 5 x Exploration Simultaneously graph = = Y x Y x , , 1 2 3 and = Y x 3 3 on a square screen with − ≤ ≤ x 3 3. What do you observe about the graphs of = Y x , 2 3 its inverse = Y x 3 3 , and the line = Y x? 1 Repeat this experiment by simultaneously graphing = Y x, 1 = + Y x2 3, 2 and = − Y x 3 2 3 on a square screen with − ≤ ≤ x 6 3. Do you see the symmetry of the graph of Y2 and its inverse Y3 with respect to the line = Y x? 1
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