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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

Understanding the function

The function given is \(f(x) = \frac{x^{3}}{2}\). It is a cubic function which means graphically, it will have a unique curve shape.
02

Graph the function

Use a graphing utility to plot the graph of \(f(x) = \frac{x^{3}}{2}\). The graph will be a curve that starts from the bottom left, crosses the origin (0,0) and heads upwards to the right. The function is increasing on the whole real line.
03

Apply the horizontal line test

The horizontal line test is used to determine if a function is one-to-one. If a horizontal line intersects the graph of the function at most one time, the function is one-to-one. In the case of \(f(x) = \frac{x^{3}}{2}\), any horizontal line drawn will intersect the graph at only one point. This means the function passes the horizontal line test and is one-to-one.
04

Determine if the function has an inverse

Since the function \(f(x) = \frac{x^{3}}{2}\) is one-to-one, we can say that it has an inverse that is also a function. This is based on the property that a function is invertible if and only if it is one-to-one.

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Most popular questions from this chapter

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