Problem 82 - 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). $$f(x)=(x-1)^{3}$$

Step-by-Step Solution

Answer

Yes, the function \(f(x)=(x-1)^3\) is one-to-one, thus it has an inverse that is also a function.

Step 1: Graphing the Function

This exercise requires the use of a graphing utility to depict the function \(f(x)=(x-1)^3\). The graph will show a curve that rises from the bottom left, makes a turn at the point (1,0) and continues to the top right of the graph. This appearance is typical of a cubic function where the highest degree of 'x' is 3.

Step 2: Application of the Horizontal Line Test

To ascertain whether a function is one-to-one and hence has an inverse that is a function, apply the horizontal line test on the graph. A horizontal line drawn through any point on the graph of the function should touch the graph at only one point if the function is to have an inverse that is also a function. For the function \(f(x)=(x-1)^3\), every horizontal line will intersect the graph at one point only, verifying that the function is one-to-one.

Step 3: Conclusion

Since the function \(f(x)=(x-1)^3\) passes the horizontal line test, it can be concluded that this function is one-to-one. Consequently, it does have an inverse that is also a function.