Problem 78 - College Algebra Essentials

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 therefore has an inverse that is also a function.

Step 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.