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Consider the function f(x) = (x+1)2. Sketch the graph of f(x) and use the horizontal line test to show that f(x) is not one-to-one. Show that f(x) is one-to-one on the restricted domain [−1,∞). Determine the domain and range for the inverse of f(x) on this restricted domain and find a formula for f−1(x).

Solution Preview

The graph of f is the graph of y=x2 shifted left 1 unit. Since there exists a horizontal line intersecting the graph more than once, f is not one-to-one. On the interval [−1,∞), f is one-to-one. The domain and range of f-1 are given by the range and domain of f, respectively. Therefore, the domain of f-1 is [0,∞) and the range of f-1 is [−1,∞).  To find a formula for f-1, solve the equation, y = (x+1)2 for x.  If y = (x+1)2, then x = −1 ± √y.  Since we are restricting the domain to the interval where x≥−1, we need ±√y≥0.  Therefore, x=−1+√y.  Interchanging x and y, we write y=−1+√x and conclude that f-1(x) = −1+√x.

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