Problem 53 - 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). $$f(x)=\sqrt[3]{2-x}$$

Step-by-Step Solution

Answer

By graphing and observing the function \(f(x)=\sqrt[3]{2-x}\), and by applying the horizontal line test, it can be determined if the function has an inverse function or not. The successful completion of the horizontal line test shows that the function is one-to-one, thus it has an inverse function.

Step 1: Graph the function

Use the graphing utility to plot the cubic root function \(f(x)=\sqrt[3]{2-x}\).

Step 2: Evaluate the graph

Observe the plotted graph. If the curve intersects vertical lines more than once, then the function is not one-to-one. Recall that if a horizontal line intersects the function's graph in more than one point, then the function does not have an inverse. This is known as the horizontal line test.

Step 3: Determine if the function has an inverse

Based on the plot of the function, if the horizontal line test is successful (if no horizontal line intersects the graph more than once), it can be concluded that the function is one-to-one and therefore, it does have an inverse.