Problem 57 - 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. $$f(x)=\frac{2 x}{x-1}$$

Step-by-Step Solution

Answer

The inverse of the given function is \(f^{-1}(x) = \frac{x}{x-2}\). When both the function and its inverse are graphed on the same set of axes, they appear as two hyperbolas, reflected about the line \(y = x\).

Step 1: Find the inverse of the function

To find the inverse of a function \(f(x) = y\), the first step is to switch the positions of \(x\) and \(y\). Thus, we get \(x = \frac{2y}{y-1}\). The next step is to solve this equation for \(y\). Firstly, cross-multiply to eliminate the fraction: \(x(y-1) = 2y\), this simplifies to \(xy-x=2y\). Secondly, we isolate the \(y\)-terms on one side, which gives us \(xy-2y = x\). Then, factor out \(y\) from the left side to get \(y(x-2) = x\). Finally, divide both sides by \((x-2)\), to solve for \(y\), which is the inverse function. Hence, the inverse function \(f^{-1}(x) = \frac{x}{x-2}\).

Step 2: Graph the function and its inverse

To graph the function and its inverse, first make a table of values for the function \(f(x) = \frac{2x}{x-1}\) and its inverse \(f^{-1}(x) = \frac{x}{x-2}\). After this, draw the Cartesian plane and plot these values. The function \(f(x)\) will be a hyperbola that opens to the right and the left. The function \(f^{-1}(x)\) will also be a hyperbola, but it will be reflected across the line \(y = x\) since that's how a function and its inverse relate graphically. Make sure that both graphs are on the same set of axes for comparison.

Key Concepts

graphing functions

Graphing functions is all about visualizing mathematical relationships on a coordinate plane. In this context, it helps us understand how input values (\( x \)) map to output values (\( y \)) for a given formula or equation. Creating a graph involves several key steps:

• Create a set of values: Choose a selection of \( x \) values and calculate their corresponding \( y \) values using the function equation.
• Plot the points: These \( x,y \) pairs are plotted as points on the Cartesian plane.
• Connect the points: Draw lines or curves through these points to visualize the function.

For the function \( f(x) = \frac{2x}{x-1} \) and its inverse \( f^{-1}(x) = \frac{x}{x-2} \), plotting these will help us see their behavior and relationship. The graph of the inverse also shows how the function reflects across the line \( y = x \).

hyperbolas

A hyperbola is a type of curve on the graph described by certain types of functions, such as rational functions. For the original function \( f(x) = \frac{2x}{x-1} \), it forms a hyperbola.
The distinctive feature of hyperbolas is their two separate branches that curve in opposite directions. They often have asymptotes, which are lines the hyperbola approaches but never touches. This happens because the function involves division by a variable, which leads to restrictions in the domain.
To understand hyperbolas better:

• They are usually shaped like an "X" or have two loops facing away from each other.
• They have asymptotes that act as invisible boundaries for the curve.
• Their position and shape change depending on the formula used.

For \( f(x) \) and \( f^{-1}(x) \), each has its own hyperbola design due to their unique equations, and one will be the mirror image of the other around \( y = x \).

Cartesian plane

The Cartesian plane is a flat, two-dimensional surface where we graph our functions. It's formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Together, they create four quadrants where points are plotted using \( (x,y) \) coordinates.
To effectively use the Cartesian plane:

• Understand that each point is defined by an \( x \) (\( horizontal \)) and a \( y \) (\( vertical \)) value.
• The origin, where \( x = 0 \) and \( y = 0 \) meet, is the center point.
• Graphs can extend infinitely, covering positive and negative values.

When graphing both \( f(x) = \frac{2x}{x-1} \) and its inverse \( f^{-1}(x) = \frac{x}{x-2} \) on the same set of axes, the Cartesian plane allows us to spatially compare their behaviors. This dual plotting can reveal symmetries and inversions—key insights when studying inverse functions.