Question

Find the inverse function of \(f\). Graph (by hand) \(f\) and \(f^{-1}\). Describe the relationship between the graphs. $$ f(x)=x^{5} $$

Short answer

The inverse function of \(f(x) = x^{5}\) is \(f^{-1}(x) = x^{1/5}\). The graphs of \(f(x)\) and \(f^{-1}(x)\) are mirror images of each other across the line \(y = x\).

Step-by-step solution

Calculate the Inverse

To get the inverse of \(f(x)\), we substitute \(f(x)\) with \(y\), exchange the places of \(y\) and \(x\) and then solve for \(y\). This gives us \(x = y^{5}\). Taking the fifth root of both sides to solve for \(y\), we find the inverse function \(f^{-1}(x) = x^{1/5}\).

Graph the Function and Its Inverse

To graph these functions by hand, mark points on the x and y axes to guide sketching the curve. For \(f(x)\), choose -2, -1, 0, 1, 2 for the \(x\)-values and plot their corresponding \(y\) values which are \(x^{5}\). For \(f^{-1}(x)\), choose -2, -1, 0, 1, 2 for \(y\) values and plot their corresponding \(x\) values which are \(y^{1/5}\). This results in smooth curves for both functions.

Describe the Relationship

The graph of \(f^{-1}(x)\) is a reflection of \(f(x)\) in the line \(y = x\). This is because inverse functions 'undo' each other and mirror each other around the line \(y = x\). So, if a point \(a, b\) lies on the graph of \(f(x)\), then its mirror point \(b, a\) lies on the graph of \(f^{-1}(x)\).

Key concepts

Graphing Functions

Graphing functions is a fundamental skill in calculus, which allows students to visualize mathematical concepts and their behaviors.
When graphing the function
\(f(x) = x^5\), the most crucial aspect is to select strategic values for \(x\) to plot the corresponding \(y\) values. It helps to start with simple integers and zero. For instance, when \(x\) is -2, -1, 0, 1, and 2, we get \(y\) values of -32, -1, 0, 1, and 32, respectively.
To do this effectively, create a small table of values, and for each chosen \(x\), calculate the \(y = x^5\). Plot these points on a coordinate plane and connect them smoothly to reveal the curvature of the graph.
The resulting graph of \(x^5\) will show that it passes through the origin and increases rapidly as \(x\) moves away from zero, giving a steep curve on both the positive and negative sides of the y-axis.

Reflecting Graphs

Reflecting graphs across a line, such as the line \(y = x\), is a visual representation of inverse functions.
To reflect a graph, imagine a mirror placed along the line of reflection. Every point on the original graph has a corresponding reflected point on the other side of the line. For the function \(f(x) = x^5\) and its inverse \(f^{-1}(x) = x^{1/5}\), plotting both on the same axes will show this property.
A point \((a, b)\) on the graph of \(f(x)\) will have a reflection point \((b, a)\) on the graph of \(f^{-1}(x)\). This reflection across the line \(y = x\) illustrates that the inverse function is doing the opposite of the original function, effectively 'undoing' the operation of the fifth power.

Fifth Root

Understanding the fifth root is essential when working with inverse functions involving fifth powers, like in our example function \(f(x) = x^5\).
The fifth root of a number \(x\) is a value \(y\) such that \(y^5 = x\). It is written as \(x^{1/5}\) or \(\sqrt[5]{x}\). When we find the inverse function of \(f(x) = x^5\), we're looking for what value, when raised to the fifth power, gives us \(x\).
The fifth root is crucial for representing the inverse function because it unravels the fifth power. By graphing \(f^{-1}(x) = x^{1/5}\), students can see how the steep curve of \(f(x)\) is smoothed out and reflected across the line \(y = x\) to create the graph of the inverse function. This graphical representation helps in visualizing the relationship between a function and its inverse.