Problem 82 - 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

Yes, the graph of the function \(f(x)=(x-1)^{3}\) indicates it is a one-to-one function, and therefore its inverse would be a function as well.

Step 1: Graph the function

Start by plotting the function \(f(x)=(x-1)^{3}\) using a graphing utility. The function is a simple cubic function shifted one unit to the right. It should produce a curve that increases for x<1, then decreases for x > 1.

Step 2: Determine whether the function is one-to-one

A function is one-to-one if any horizontal line drawn through it intersects the function exactly once. After looking at the graph of the function \(f(x)\), it can be observed that any horizontal line will intersect the curve exactly once. Therefore, the function is one-to-one.

Step 3: Conclusion

Since \(f(x)=(x-1)^{3}\) is a one-to-one function based on the horizontal line test, it means its inverse is also a function.

Key Concepts

Inverse Functions

Inverse functions are fascinating because they essentially reverse the roles of the input and the output of the original function. If you have a function \( f(x) \), its inverse, denoted as \( f^{-1}(x) \), will swap the roles of \( x \) and \( f(x) \). This means that if \( f(a) = b \), then \( f^{-1}(b) = a \).

To find if a function has an inverse that is also a function, the original function must be one-to-one. This implies that each output is produced by exactly one input value. An inverse can only exist in this case because otherwise, we'd have ambiguity where two inputs could produce the same output, confusing our reverse process. It ensures that the inverse process is unambiguous.

• One-to-one functions pass the horizontal line test (more on that below).
• If a function is one-to-one, its inverse is guaranteed to be a function as well.
• The graph of the inverse is symmetrical to the graph of the original function with respect to the line \( y = x \).

Horizontal Line Test

The horizontal line test is a simple yet powerful tool to determine if a function is one-to-one. When we say a function is one-to-one, we're saying that for every second input argument, it maps to one distinct output. This is crucial because only one-to-one functions have inverses that are well-defined functions.

Here's how the test works:

• Draw or visualize horizontal lines across the graph of the function.
• Observe where these lines intersect the graph.
• If any horizontal line crosses the graph more than once, the function is not one-to-one.
• If each line crosses only one point, the function is indeed one-to-one.

For our function \( f(x) = (x-1)^3 \), any horizontal line will intersect the curve exactly once, showing that it is a one-to-one function. This also confirms that you can find an inverse for this function.

Graphing Utility

Graphing utilities are indispensable tools for visualizing mathematical functions and their properties. They allow you to plot a function almost instantly, providing clarity that helps ascertain various properties of the function, such as whether it is one-to-one.

To graph the function \( f(x) = (x-1)^3 \) using a graphing utility:

• Input the function into the graphing software.
• Observe how the curve behaves across different values of \( x \).
• Use the graph to perform a horizontal line test by imagining or sketching horizontal lines across the output.
• Analyze the intersection points as outlined in previous sections.

This visual approach makes it easy for students to confirm the one-to-one nature of a function, making the graphing utility a core component in understanding complex mathematical concepts.