Problem 52 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=\frac{-1}{2 x}$$

Step-by-Step Solution

Answer

The inverse of the function \(g(x) = \frac{-1}{2x}\) is \(g^{-1}(x) = \frac{-1}{2x}\). When graphed on the same set of axes, the graph of \(g(x)\) and its inverse are symmetric with respect to the line \(y=x\).

Step 1: Find the Inverse of the Function

To find the inverse of the function \(g(x) = \frac{-1}{2x}\), interchange \(x\) and \(y\), which gives \(x = \frac{-1}{2y}\). Now, solve for \(y\) to get the inverse function. Multiply both sides by \(2y\) to get \(2xy = -1\) and then divide both sides by \(2x\) to get \(y = \frac{-1}{2x}\), which is the inverse function.

Step 2: Graph the Original Function and Its Inverse

To graph \(g(x) = \frac{-1}{2x}\) and its inverse on the same set of axes, firstly, draw the graph of \(g(x)\). As \(x\) approaches any positive or negative large number, \(y\) approaches 0. And as \(x\) approaches 0 from the positive side, \(y\) approaches \(-\infty\), and from the negative side, \(y\) approaches \(+\infty\). Now, plot the graph of the inverse function, which will be reflected about the line \(y=x\).

Step 3: Draw the Line \(y=x\)

The line \(y = x\) is the line of symmetry between a function and its inverse. Therefore, draw this line on the same axes to visually confirm that the function \(g(x)\) and its inverse are symmetric with respect to the line \(y=x\).

Key Concepts

Function Graphing

Graphing functions is a fundamental way to understand their behavior visually. For the function \( g(x) = \frac{-1}{2x} \), its graph will give us a clear idea of how it behaves for different values of \( x \). To graph a function like this, follow these steps:

• Identify the function's behavior as \( x \) approaches different limits; here, we see that as \( x \) increases or decreases significantly, \( g(x) \) approaches zero.
• Plot a few specific values to get a set of anchor points. For instance, if you choose certain \( x \) values, calculate the corresponding \( g(x) \) and mark these points.
• Note that as \( x \) approaches zero from either direction, \( g(x) \) tends to infinity in both the negative and positive directions. This gives the graph a hyperbolic shape.

Graphing not only shows us the function's general shape but helps us analyze the changes in its rate and direction as it curves and approaches axes.

Functional Symmetry

Functional symmetry is an important concept when dealing with inverse functions. When you look at the original function \( g(x) = \frac{-1}{2x} \), and its inverse, you’ll see that they have reflective symmetry about the line \( y = x \).

• The reflective symmetry means that if you were to fold the plane along the line \( y = x \), the function and its inverse would align perfectly.
• This symmetry indicates that if \((a, b)\) is a point on the original function’s graph, then its inverse will reflect this as a \((b, a)\) point.

Understanding how functions and their inverses relate through symmetry helps to deepen comprehension of their graphical behavior and how they transition from one to another on a plot.

Lines of Symmetry

Lines of symmetry serve as a visual tool to better understand the relationship between functions and their inverses. In the context of the provided exercise, the line of symmetry is \( y = x \).

• This line is pivotal for checking if two functions are indeed inverses since they should mirror each other across it.
• Graphically, after drawing both the function \( g(x) = \frac{-1}{2x} \) and its inverse, plot the line \( y = x \). Ensure both graphs reflect symmetrically across this line. If you notice matching points on either side of this line for corresponding values, you've confirmed their inverse nature.
• Lines of symmetry help in understanding how inverses ‘swap’ the coordinates of points. If you visualize this on the graph, it supports the algebraic steps of finding and confirming inverses.

Including lines of symmetry in graph plotting enables a complete understanding of function behaviors and the transition to their inverses.