Problem 37 - Calculus of a Single Variable

Question

Find the inverse function of \(f,(\mathbf{b})\) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, ( \(\mathbf{c} )\) describe the relationship between the graphs, and ( \(\mathbf{d} )\) state the domain and range of \(f\) and \(f^{-1} .\) \(f(x)=x^{5}\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^{5}\) is \(f^{-1}(x) = \sqrt[5]{x}\). The graphs of \(f(x)\) and \(f^{-1}(x)\) are reflections of each other over the line \(y = x\). Both functions have a domain and range of all real numbers.

Step 1: Finding the inverse function

The first step is to find the inverse function of \(f(x)\). This can be done by interchanging \(y\) and \(x\) in the equation and then solving for \(y\). So, our function is \(f(x) = x^{5}\). Replace \(f(x)\) with \(y\), so the function becomes \(y = x^{5}\). By interchanging \(x\) and \(y\), the equation becomes \(x = y^{5}\). Solving for \(y\) gives \(y = \sqrt[5]{x}\). Thus, the inverse function is \(f^{-1}(x) = \sqrt[5]{x}\).

Step 2: Graphing the functions

Now, you need to graph the original function \(f(x) = x^{5}\) and its inverse function \(f^{-1}(x) = \sqrt[5]{x}\) on the same set of axes. Remember that the graphs of \(f\) and \(f^{-1}\) are mirror images of each other over the line \(y = x\). Using a graphing calculator or a graphing software tool can help you sketch the functions more accurately.

Step 3: Describing the relationship

From the graph, one can see that the original function \(f(x) = x^{5}\) and its inverse \(f^{-1}(x) = \sqrt[5]{x}\) are reflections of each other over the line \(y = x\). This is a typical property of a function and its inverse.

Step 4: State the domain and range

The domain of a function is the set of all possible values of \(x\) for which the function is defined, and the range is the set of all possible values of \(y\). For the original function \(f(x) = x^{5}\), the domain is all real numbers because \(x\) can be any real number, and the range is also all real numbers, because a power of \(5\) can yield any real number. For the inverse function \(f^{-1}(x) = \sqrt[5]{x}\), the domain is also all real numbers since the fifth root of any real number is defined. The range is similarly all real numbers.

Key Concepts

Finding Inverse Functions

Understanding how to find inverse functions is a fundamental concept in calculus. When we have a function like \(f(x) = x^{5}\), we might want to know which function will 'undo' this operation. To find the inverse function, we typically follow a series of steps. First, we replace \(f(x)\) with \(y\) to write the function as \(y = x^{5}\). Then we switch \(x\) and \(y\), leading to \(x = y^{5}\). Now, our task is to solve for \(y\), which gives us the inverse function, \(f^{-1}(x) = \(\sqrt[5]{x}\)\).

To see if we've found the correct inverse, we can apply a quick check. Composing the original function with its inverse should yield the identity function. In other words, applying \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) both should give us \(x\). This step is crucial for ensuring that the inverse function correctly reverses the action of the original function.

Graphing Functions

Graphing functions and their inverses provides a visual representation of their relationship. For \(f(x) = x^{5}\) and its inverse \(f^{-1}(x) = \(\sqrt[5]{x}\)\), graphing these on coordinate axes helps us to see the symmetry between them. This symmetry occurs across the line \(y = x\), which acts as a mirror dividing the two functions.

When graphing, ensure to plot multiple points for a clear depiction. Remember that the more complex the function, the more points you'll need to plot to capture its shape accurately. The use of graphing technology can be extremely helpful here, allowing for precise curves that represent the functions. For students, this is an excellent way to verify if the inverse function has been determined correctly; if it reflects accordingly across the line \(y = x\), then the inverse is likely correct.

Domain and Range of Functions

The domain of a function refers to all the possible input values, and the range refers to all the possible output values. For the function \(f(x) = x^{5}\), the domain and range include all real numbers; simply, any real number raised to the fifth power is still a real number.

Similarly, its inverse \(f^{-1}(x) = \(\sqrt[5]{x}\)\) also has all real numbers as its domain and range since the fifth root of any real number is defined. It's essential to remember that the domain and range of an inverse function are interchanged compared to those of the original function. This interchanging of domain and range is another characteristic property of inverse functions, reinforcing the deep connection between a function and its inverse.