Problem 42 - College Algebra

Question

For the following exercises, find the inverse function. Then, graph the function and its inverse. $$ f(x)=\frac{3}{x-2} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{3 + 2x}{x} \).

Step 1: Define the Function

The given function is \( f(x) = \frac{3}{x-2} \). To find the inverse, we start by replacing \( f(x) \) with \( y \), giving us \( y = \frac{3}{x-2} \).

Step 2: Swap Variables

To find the inverse, swap \( x \) and \( y \) in the equation: \( x = \frac{3}{y-2} \). This will help us solve for \( y \), which is the inverse function.

Step 3: Solve for y

To solve \( x = \frac{3}{y-2} \) for \( y \), first multiply both sides by \( y-2 \), resulting in \( x(y-2) = 3 \).

Step 4: Simplify Equation

Distribute \( x \) on the left side: \( xy - 2x = 3 \).

Step 5: Solve for y

Rearrange the equation to isolate \( y \): \( xy = 3 + 2x \).

Step 6: Finalize y

Divide both sides by \( x \) to get \( y = \frac{3 + 2x}{x} \).

Step 7: Write the Inverse Function

The inverse function is \( f^{-1}(x) = \frac{3 + 2x}{x} \).

Step 8: Graph the Functions

Graph \( f(x) = \frac{3}{x-2} \) and its inverse \( f^{-1}(x) = \frac{3 + 2x}{x} \). These functions should be symmetric about the line \( y = x \).

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in mathematics that allows you to visualize the behavior of functions. To plot a function like \( f(x) = \frac{3}{x-2} \), we start by identifying the basic properties: the domain, range, asymptotes, and intercepts. Since dividing by zero is undefined, the domain of this function is all real numbers except 2. You will also notice a vertical asymptote at \( x = 2 \), where the function behaves asymptotically, shooting towards positive or negative infinity.
To graph this function, it's helpful to plot key points on each interval divided by the asymptote. For instance, examining values such as \( x = 1, x = 0, x = -1 \) for \( x < 2 \) and values like \( x = 3, x = 4, x = 5 \) for \( x > 2 \) will depict how the function changes between these points.

• The function \( f(x) \) never touches the asymptote acknowledging its boundary behavior.
• Look for intercepts for easier plotting, like finding where the function cuts through the axes, which helps in rendering an accurate graph.

Graphing the inverse function \( f^{-1}(x) = \frac{3 + 2x}{x} \) follows a similar process but remember that its domain will differ, emphasizing the distinctions between the original and inverse functions.

Function Transformation

When talking about function transformation, it refers to changing the function's position, shape, or orientation on the graph. Function transformations can be broken down into a few key types: translations, reflections, stretches, and compressions.
Let’s consider the function \( f(x) = \frac{3}{x-2} \). Translating involves adjusting the function horizontally or vertically. The term "\(-2\)" in \( x-2 \) indicates a horizontal translation to the right by 2 units. So, every point on the graph of \( f(x) = \frac{3}{x} \) moves rightward.

• Vertical and horizontal shifts are very intuitive. Moving a function up or down, left or right refers to simply moving or shifting the graph without altering its shape.
• A reflection could be seen if we multiplied the function by \(-1\) which would flip it over an axis.
• Stretching the graph could mean multiplying all of the \( y \)-coordinates by \( c > 1 \), making the graph taller.

By graphing transformations, mathematicians can predict and understand behavioral changes in functions efficiently. Understanding transformations is essential for manipulating and graphing more complex functions.

Symmetry in Algebra

Symmetry in algebra refers to a function’s graph exhibiting a mirror-like or symmetric property. One way symmetry is observed is through inverse functions. When two functions \( f(x) \) and \( f^{-1}(x) \) are graphed, they often display symmetry about the line \( y = x \). This is crucial because it visually demonstrates the inverse relationship.
In the exercise, when graphing \( f(x) = \frac{3}{x-2} \) and its inverse \( f^{-1}(x) = \frac{3 + 2x}{x} \), one can observe that flipping across the line \( y = x \) reveals their inverse relationship with mirrored properties manifesting in their graphical layouts.

• Symmetry gives insight into balanced and unbalanced aspects of functions, showcasing consistent patterns.
• Recognizing this symmetry in functions helps to easily understand and predict function behavior.
• Exploiting symmetry simplifies complex calculations and problem-solving by reducing redundancy.

Understanding symmetry allows algebra students to streamline equations and predictions about function behavior that save on computation time or errors in graphing.