Problem 39 - College Algebra

Question

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

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{2}{x} \). Both the function and its inverse are the same.

Step 1: Write the function as an equation

To find the inverse, start by writing the function as an equation: \( y = \frac{2}{x} \). In this step, we identify that \( y \) is the output of the function for a given input \( x \).

Step 2: Swap the variables

To find the inverse, swap the \( x \) and \( y \) in the equation to reflect the function's output as the input of the inverse function. This gives us \( x = \frac{2}{y} \).

Step 3: Solve for the new 'output'

Now, solve the equation \( x = \frac{2}{y} \) for \( y \). Multiply both sides by \( y \) to get \( xy = 2 \).

Step 4: Rearrange to express y in terms of x

To isolate \( y \), divide both sides by \( x \). This gives the inverse function as \( y = \frac{2}{x} \).

Step 5: Graph the functions

Graph the original function \( f(x) = \frac{2}{x} \) and its inverse \( f^{-1}(x) = \frac{2}{x} \). Notice that in this case, the function and its inverse are identical, and both are hyperbolas with the line \( y = x \) serving as the line of symmetry.

Key Concepts

Understanding Function Graphing

Graphing is an essential skill in mathematics that enables us to visualize how functions behave. When we graph a function like \( f(x) = \frac{2}{x} \), we're essentially plotting the set of coordinates \((x, y)\) where \( y \) is calculated using the function.
Graphing helps us understand the unique characteristics of functions, such as their lines of symmetry, intercepts, and asymptotes.

• For \( f(x) = \frac{2}{x} \), the graph creates a hyperbola, a specific type of curve.
• The hyperbola has two branches, one in the first quadrant and one in the third.
• Each branch approaches the x-axis and y-axis but never actually touches them. These are called asymptotes.

This visually distinctive profile of hyperbolas helps identify inverse relations once plotted on a graph. You should always remember that when graphing a function and its inverse, these graphs will often mirror each other across the line \( y = x \). This offers a powerful way not only to see the correlation between the function and its inverse but also to check the accuracy of your work.

The Role of Variable Swapping

Swapping variables is a fundamental step in finding the inverse of a function. It illustrates how the input and output of functions are interchanged in their inverses. After writing the function \( y = \frac{2}{x} \), you swap \( x \) and \( y \) to get \( x = \frac{2}{y} \).
This operation transforms the perspective: the output of the original becomes the input of the inverse, and vice versa.

• This switching shows that every solution pair \((x, y)\) in the original function becomes \((y, x)\) in the inverse.
• It's crucial as it reveals the inverse function's plot, where you essentially 'flip' the points over the line \( y = x \).

By understanding this swapping, you become adept at deducing inverses of various functions. This swapping aids in visualizing the reflective symmetry present when plotting both a function and its inverse graphically.

Hyperbola Symmetry and Its Unique Properties

In mathematics, a hyperbola is a type of smooth curve lying in a plane. The function \( f(x) = \frac{2}{x} \) generates a hyperbolic graph, characterized by its mirror symmetry. This symmetry is a crucial element, especially when discussing the function and its inverse.
For hyperbolas:

• The graph of \( f(x) = \frac{2}{x} \) has reflection symmetry related to both the x-axis and y-axis.
• Consider that the line \( y = x \) acts as the line of symmetry between the function and its inverse.

Both the function \( f(x) \) and its inverse \( f^{-1}(x) \) have the same hyperbolic form, reflecting across \( y = x \) itself. This property provides a visual confirmation that they are indeed inverses. Thanks to this symmetry, recognizing and verifying hyperbolic functions becomes more straightforward, allowing for greater ease in not only solving but also understanding mathematical problems involving hyperbolas.