Problem 79 - 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. $$f(x)=\frac{4}{x^{3}}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \frac{4}{x^3}\) is \(f^{-1}(x) = \left(\frac{4}{x}\right)^{1/3}\). When both \(f(x)\) and \(f^{-1}(x)\) are graphed in the same viewing window, one should observe symmetry about the line \(y = x\), suggesting the functions are inverses of each other.

Step 1: Find the inverse function of \(f(x)\)

First, replace \(f(x)\) with \(y\) to get \(y = \frac{4}{x^3}\). Swap \(x\) and \(y\) to obtain \(x = \frac{4}{y^3}\). Solving for \(y\) gives the inverse function \(f^{-1}(x) = \left(\frac{4}{x}\right)^{1/3}\).

Step 2: Graph the original and inverse function

Use a graphing utility to graph \(f(x) = \frac{4}{x^3}\) and its inverse \(f^{-1}(x) = \left(\frac{4}{x}\right)^{1/3}\). As a guide, the function \(f(x) = \frac{4}{x^3}\) can be graphed as a hyperbola opening along the axes, while the graph of the inverse function \(f^{-1}(x) = \left(\frac{4}{x}\right)^{1/3}\) will be its reflection along the line \(y = x\).

Step 3: Analyze the graph

By viewing both graphs in the same window, the symmetry about the line \(y = x\) should be observable. This shows the relationship between a function and its inverse, which is consistent with the analytical solution derived in Step 1.

Key Concepts

Inverse Functions

The concept of inverse functions is a fundamental aspect of algebra that deals with reversing the effect of a given function. When we have a function, denoted by \(f(x)\), finding its inverse, denoted by \(f^{-1}(x)\), essentially means we are looking for a function that 'undoes' what the original function does.

For example, if you have a function that squares a number, its inverse function would take the square root of that number. To find the inverse function algebraically, we follow three main steps:

• Replace the function notation \(f(x)\) with \(y\).
• Switch the roles of \(x\) and \(y\) in the equation.
• Solve for \(y\), which will give you the inverse function \(f^{-1}(x)\).

These steps transform the given function into its inverse. In the provided exercise, the function \(f(x)=\frac{4}{x^{3}}\) was transformed into \(f^{-1}(x) = \left(\frac{4}{x}\right)^{1/3}\) following these steps. It's crucial to remember that not all functions have inverses, particularly those that are not one-to-one; these are functions where each value of \(x\) produces a unique value of \(y\).

Graphing Functions

Graphing functions is a visual way of understanding mathematical relationships and behaviors. A graph represents all the possible values a function can take and gives us a clear picture of its nature. When graphing a function such as \(f(x)=\frac{4}{x^{3}}\), which was given in the exercise, we plot all the points \((x, y)\) where \(y\) is the output of the function when \(x\) is the input.

For the function in the exercise, the graph takes the shape of a hyperbola, which curves away from the axes in two directions. To represent this on graph paper or using a graphing utility, we would plot several points based on values we choose for \(x\), and then connect those points to form a continuous curve.

Improving the Grasping of Graphing Concepts

When learning about graphing, it's beneficial to experiment with various functions to see how changes in the equation affect the graph's shape. Identifying patterns in these changes helps deepen understanding of function behaviors. Additionally, learning how to graph functions effectively is incredibly useful for visual comparisons, such as comparing a function to its inverse.

Symmetry in Graphs

Symmetry in graphs is an intriguing feature that often indicates a special relationship between a function and its inverse. In general, two types of symmetry are talked about in mathematics: symmetry with respect to an axis and symmetry with respect to a point. In the context of inverse functions, we are most interested in symmetry with respect to the line \(y = x\).

When you graph a function and its inverse on the same coordinate plane, they should be mirror images of each other across this line. This means that every point \((x, y)\) on the graph of the function has a corresponding point \((y, x)\) on the graph of the inverse function.

Visualizing Symmetry Between Function and Inverse

It's essential to employ graphing skills to see this relationship. Using a graphing utility, as in the exercise, helps students verify the accuracy of their algebraic solutions in finding inverses. Seeing both graphs side by side reinforces the concept that operations carried out by a function are 'reversed' by its inverse. Understanding symmetry can also support comprehension when studying more complex functions or when solving equations graphically.