Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) algebraically. Use a graphing utility to graph both \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$f(x)=x^{5}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^{5}\) is \(f^{-1}(x) = \sqrt[5]{x}\).

Step 1: Determine the inverse function

The inverse function is found by swapping \(x\) and \(y\) and then solving the equation for \(y\). Here, replace \(f(x)\) with \(y\), i.e. \(y = x^{5}\). Now, swap \(x\) and \(y\) to obtain \(x = y^{5}\). To solve for \(y\), take the fifth root on both sides to obtain \(y = \sqrt[5]{x}\). Thus, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = \sqrt[5]{x}\).

Step 2: Verify the Inverse Function

For any function and its inverse, \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Substitute \(f^{-1}(x)\) into \(f(x)\) and check if you get \(x\) as the result. Here, \(f(f^{-1}(x)) = (\sqrt[5]{x})^{5} = x\) and likewise, \(f^{-1}(f(x)) = \sqrt[5]{(x^{5})} = x\). Thus, it is verified that the calculated inverse function is correct.

Key Concepts

Algebraic Operations

When it comes to inverse functions in algebra, understanding the importance of algebraic operations is crucial. Inverse functions essentially reverse the effect of the original function. To find an inverse function algebraically, we often perform a series of operations in a precise order.

To start, we replace the function notation with a variable, usually swapping out the function's output notation, for example, replacing \(f(x)\) with \(y\). Then, we switch the roles of \(x\) and \(y\) to reflect the inverse relationship. This step is essential as it frames the problem in a way that allows us to isolate the new \(y\), which will become the inverse function. In the case of \(f(x) = x^{5}\), we swap to get \(x = y^{5}\) and then perform inverse operations. Since \(x^{5}\) involves an exponent of 5, the inverse operation would be the fifth root. Thus, applying the inverse operation, we get the inverse function \(f^{-1}(x) = \root5\right)\backslash x\).

Graphing Utility

A graphing utility is an essential tool for visualizing the relationship between a function and its inverse. It becomes particularly handy when dealing with complex functions where the inverse may not be readily apparent.

After finding the inverse algebraically, plotting both functions on the same graph can reveal their connection. For instance, when you graph the function \(f(x) = x^{5}\) and its inverse \(f^{-1}(x) = \root5\right)\backslash x\), you will notice that they are reflections of each other about the line \(y = x\). This is a defining characteristic of functions and their inverses. The mirroring effect shown by a graphing utility is a visual confirmation that you have found the correct inverse.

Function Verification

After obtaining the inverse function, it's imperative not to skip the step of verification. Verifying that your inverse function actually 'undoes' the original function is how you ensure the correctness of your work.

To verify, we use the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). For a function and its inverse, both compositions should yield \(x\). If you substitute your inverse function into the original function, the algebraic operations should cancel out and leave you with \(x\), confirming that you have indeed found the true inverse. In our example, plugging in \(f^{-1}(x) = \root5\right)\backslash x\) into \(f(x)\) and vice versa indeed resulted in \(x\), thus validating our inverse function.