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Chapter 2: Problem 82

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=(x-1)^{3} $$

Short Answer

Expert verified
The function \(f(x)=(x-1)^{3}\) is one-to-one and hence it has an inverse that is also a function.

Step by step solution

01

Graph the function

Use a graphing utility to plot the function \(f(x)=(x-1)^{3}\). Observe how the function behaves and look for possible multiple intersections with hypothetical horizontal lines.
02

Apply the horizontal line test

If a horizontal line crosses the function graph at more than one point, then the function is not one-to-one. However, from observing the graph, you can see that no horizontal line crosses the graph more than once. Therefore, the function passes the horizontal line test.
03

Determine if the function has an inverse

Since the function passes the horizontal line test, it is one-to-one and it does have an inverse that is also a function.

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