Problem 58 - 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). $$f(x)=(x-1)^{3}$$

Step-by-Step Solution

Answer

The function \(f(x) = (x-1)^3\) is one-to-one, and therefore, it has an inverse that is also a function.

Step 1: Graph the function

Start by plotting the function \(f(x) = (x-1)^3\). This is a basic cubic function shifted one unit to the right. Plot the function using a graphing utility. This will give a visual representation of the function.

Step 2: Analyze the graph

Observe the graph to see if any horizontal line intersects the function at more than one point. If it does, the function is not one-to-one. If each horizontal line intersects the function at most once, the function is one-to-one. This is called the Horizontal Line Test.

Step 3: Conclusion

If no horizontal line intersects the graphed function at more than one point, we can say that the function \(f(x) = (x-1)^3\) is indeed one-to-one, and thus, it has an inverse that is also a function.