Problem 55 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.