Problem 54 - 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). $$f(x)=\frac{x^{3}}{2}$$

Step-by-Step Solution

Answer

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

Step 1: Graph the function

Using a graphing utility, plot the function \( f(x)=\frac{x^{3}}{2} \). You will find that it creates a curving line, starting at negative infinity, passing through the origin, and proceeding towards positive infinity. The graph is symmetric with respect to the origin.

Step 2: Conduct the horizontal line test

Draw horizontal lines across the graph. If each horizontal line intersects the graph at only one point, that verifies the existence of an inverse of the function that is also a function. Looking at the graph of our function \( f(x)=\frac{x^{3}}{2} \), we can see that each horizontal line drawn through the graph will intersect at only one point.

Step 3: Conclude about the invertibility of the function

Given that the function passes the horizontal line test, we can conclude that the function is one-to-one, which means it has an inverse that is also a function.