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Chapter 2: Problem 78

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). $$ f(x)=\frac{x^{3}}{2} $$

Short Answer

Expert verified
Yes, the function \(f(x)=\frac{x^{3}}{2}\) is one-to-one and therefore has an inverse that is also a function.

Step by step solution

01

Graph the Function

Using a graphing utility, plot the function \(f(x)=\frac{x^{3}}{2}\).
02

Apply the Horizontal Line Test

To check if the function is one-to-one and validate if an inverse exists, a horizontal line test is performed. If any horizontal line intersects the graph more than once, the function is not one-to-one. In this case, draw horizontal lines across the graph.
03

Analysis of the Graph

Observe the intersections of horizontal lines and the graph. For any given horizontal line, it is seen that it intersects the graph at most once. This confirms that the function \(f(x)=\frac{x^{3}}{2}\) is a one-to-one function.

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