Question

In Exercises 45 and \(46,\) find \(f^{-1}\) over the indicated interval. Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=\frac{x}{x^{2}-4} ; \quad(-2,2) $$

Short answer

The inverse of \(f(x)=\frac{x}{x^{2}-4}\) defined over \(-2<x<2\) is \(f^{-1}(x)=\frac{2x}{1 + \sqrt{1+4x^{2}}}\). The graph of \(f(x)\) over this interval and the graph of its inverse \(f^{-1}(x)\) are symmetrical with respect to the line \(y=x\).

Step-by-step solution

Find the inverse function

To find the inverse function, first, let \(y = f(x) = \frac{x}{x^{2}-4}\). Swap \(x\) and \(y\) to get \(x = \frac{y}{y^{2}-4}\). Now solve this equation for \(y\) to get the inverse. This results in \(y = f^{-1}(x) = \frac{2x}{1 \pm \sqrt{1+4x^{2}}}\). We choose the positive square root in the denominator because \(f^{-1}(x)\) must be increasing for \(-2<x<2\).

Graph \(f(x)\) and \(f^{-1}(x)\)

Using a graphing utility, plot \(f(x) = \frac{x}{x^{2}-4}\) and \(f^{-1}(x) = \frac{2x}{1 + \sqrt{1+4x^{2}}}\) over the interval \(-2 < x < 2\).

Describe the relationship between the graphs

Observe the graphs. The graph of \(f^{-1}(x)\) is a reflection of the graph of \(f(x)\) over the line \(y=x\). Both graphs exhibit symmetry with respect to the line \(y=x\), which is a characteristic property of functions and their inverses.

Key concepts

Graphing Utility

Graphing utilities are essential tools in understanding the behavior of functions and their inverses. By allowing us to visually inspect the relationship between a function and its inverse, graphing utilities provide a hands-on approach to study the morphology of graphs.

When it comes to graphing the function \(f(x) = \frac{x}{x^2-4}\) and its inverse, particularly over an interval like \((-2,2)\), a graphing utility helps us observe the plots for both within the same window. It provides insights that might not be immediately obvious from the algebraic form alone. A reliable graphing utility will enable students to adjust the viewing window to focus on the interval of interest, ensuring that all relevant features of the graphs are visible.

Using a graphing utility not just aids in gaining visual understanding but also assists in verifying the correctness of the found inverse function. For students tackling such exercises, effective use of graphing utilities is key to cementing their understanding of inverse functions.

Reflection Property

The reflection property is a defining characteristic of function inverses. Simply put, if you graph a function and its inverse on the same axes, their graphs will be reflections of each other across the line \(y = x\).

For the function \(f(x) = \frac{x}{x^2-4}\), the step-by-step solution provided unfolds the process of finding its inverse, which when graphed together with the original function, will mirror across the identity line. This reflection property is crucial; it confirms that the operation of the function and the operation of its inverse 'undo' each other – a fundamental trait of inverse pairs. Encouraging students to look for this reflective symmetry can often verify that they've correctly found an inverse.

Function Inverses

Unlocking the concept of function inverses is vital for comprehending many mathematical principles. A function's inverse, denoted as \(f^{-1}(x)\), essentially reverses the effect of function \(f(x)\).

To find an inverse function, as shown in the provided solution, involves switching the roles of \(x\) and \(y\), and solving for \(y\) to express \(y\) as a function of \(x\). Here, for the function \(f(x) = \frac{x}{x^2-4}\), the inverse is found using algebraic manipulation and is expressed as \(f^{-1}(x) = \frac{2x}{1 + \sqrt{1+4x^2}}\) over the interval \((-2,2)\).

Teaching students to navigate this reversal process is essential for their mathematical toolkit. The ability to determine function inverses facilitates a deeper understanding of functions as they relate to their inputs and outputs.

Symmetry

Symmetry in mathematics often alludes to a balanced, harmonious arrangement of points about a particular line or point. In the realm of functions and their inverses, symmetry about the line \(y = x\) is a hallmark of this relationship.

For the function given in the exercise, \(f(x)\), and its inverse, \(f^{-1}(x)\), the graphical representation showcases this symmetry. It's a visual testament to the fact that each point on the graph of \(f(x)\) has a corresponding point on the graph of \(f^{-1}(x)\) that is equidistant from the line \(y = x\), but positioned on opposing sides.

Instructing students to verify that the symmetry is present can serve as a confirmatory check that the inverse function has been correctly identified. It's also an engaging way to appreciate the geometric beauty inherent in mathematical structures.