Chapter 1 | Functions and Graphs
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Figure 1.40 (a) For g ( x ) = x 2 restricted to [0, ∞), g −1 ( x ) = x . (b) For h ( x ) = x 2 restricted to (−∞, 0], h −1 ( x ) =− x .
Example 1.31 Restricting the Domain
Consider the function f ( x ) = ( x +1) 2 . a. Sketch the graph of f and use the horizontal line test to show that f is not one-to-one. b. Show that f is one-to-one on the restricted domain [−1, ∞). Determine the domain and range for the inverse of f on this restricted domain and find a formula for f −1 . Solution a. The graph of f is the graph of y = x 2 shifted left 1 unit. Since there exists a horizontal line intersecting the graph more than once, f is not one-to-one.
b. On the interval [−1, ∞), f is one-to-one.
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