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Chapter 2: Problem 55

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). $$f(x)=\frac{x^{4}}{4}$$

Short Answer

Expert verified
Upon graphing and applying the horizontal line test, it is observed that the function \(f(x)=\frac{x^{4}}{4}\) is not one-to-one as it fails the horizontal line test. Hence, the function \(f(x)=\frac{x^{4}}{4}\) does not have an inverse that is a function.

Step by step solution

01

Graph the Function

Using a graphing utility, graph the function \(f(x)=\frac{x^{4}}{4}\). Ensure all relevant parts of the graph are included.
02

Apply the Horizontal Line Test

Draw horizontal lines through the graph. Check if any horizontal line intersects the graph at more than one point. If a horizontal line does intersect the graph at more than one point, then the function is not one-to-one, and subsequently, it does not have an inverse that is a function. If no horizontal line intersects the graph at more than one point, the function is one-to-one and it does have an inverse that is a function. The goal of this step is to see if the graph passes the horizontal line test.
03

Analyze the Result

Based on the result of Step 2, determine whether the function has an inverse that is a function. If the graph passes the horizontal line test, confirm that the function \(f(x)=\frac{x^{4}}{4}\) is one-to-one and it does have an inverse that is a function. If the graph does not pass the horizontal line test, then function does not have an inverse that is a function.

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Most popular questions from this chapter

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