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Chapter 1 | Functions and Graphs
Solution Reflect the graph about the line y = x . The domain of f −1 is [0, ∞). The range of f −1 is [−2, ∞). Byusing the preceding strategy for finding inverse functions, we can verify that the inverse function is f −1 ( x ) = x 2 −2, as shown in the graph.
1.25 Sketch the graph of f ( x ) =2 x +3 and the graph of its inverse using the symmetry property of inverse functions.
Restricting Domains As we have seen, f ( x ) = x 2 does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of f such that the function is one-to-one. This subset is called a restricted domain . By restricting the domain of f , we can define a new function g such that the domain of g is the restricted domain of f and g ( x ) = f ( x ) for all x in the domain of g . Then we can define an inverse function for g on that domain. For example, since f ( x ) = x 2 is one-to-one on the interval [0, ∞), we can define a new function g such that the domain of g is [0, ∞) and g ( x ) = x 2 for all x in its domain. Since g is a one-to-one function, it has an inverse function, given by the formula g −1 ( x ) = x . On the other hand, the function f ( x ) = x 2 is also one-to-one on the domain (−∞, 0]. Therefore, we could also define a new function h such that the domain of h is (−∞, 0] and h ( x ) = x 2 for all x in the domain of h . Then h is a one-to-one function and must also have an inverse. Its inverse is given by the formula h −1 ( x ) =− x ( Figure 1.40 ).
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