Problem 42 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=2 x $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is a linear function, defined as \( f(x) = 2x \). Our task is to find the inverse function, denoted as \( f^{-1}(x) \).

Step 2: Switch the Variables

To find the inverse function, start by switching the roles of \( x \) and \( f(x) \). Thus, set \( y = f(x) = 2x \), and then switch \( x \) and \( y \) to get \( x = 2y \). Our goal is now to solve for \( y \).

Step 3: Solve for the Inverse

To find \( y \), rearrange the equation \( x = 2y \) to solve for \( y \): divide both sides of the equation by 2. This yields \( y = \frac{x}{2} \).

Step 4: Recognize the Inverse Function

We have solved for \( y \), which represents \( f^{-1}(x) \). Therefore, the inverse function is \( f^{-1}(x) = \frac{x}{2} \).

Key Concepts

Linear Functions

Linear functions are a core concept in algebra and mathematics. A linear function is, quite simply, a function that describes a straight line on a graph. It's defined by its equation, usually in the form of \( f(x) = ax + b \), where \( a \) is the slope, and \( b \) is the y-intercept. This means:

• The slope \( a \) indicates how steep the line is and in which direction it goes, either upwards or downwards.
• The y-intercept \( b \) tells us where the line crosses the y-axis.

For the function given in the exercise \( f(x) = 2x \), you can see that it's a simple linear function. Here, the slope \( a \) is 2 and there's no y-intercept, which means that the line passes through the origin (0,0). Linear functions like this one are important because they are foundational, ease in understanding other mathematical concepts within algebra.

Symbolic Representation

Symbolic representation in mathematics is all about expressing ideas through symbols. It's a universal language that allows us to convey complex mathematical concepts in a condensed form. For inverse functions, symbolic representation helps simplify communication by allowing us to see directly how components interact with each other.In the exercise, the function \( f(x) = 2x \) is a symbolic representation of how \( x \) relates to \( f(x) \). To find its inverse, \( f^{-1}(x) \), we switch the output with the input. This is represented symbolically by changing the variables, where \( y = f(x) \) becomes \( x = 2y \). Through solving for \( y \), you change it back to a function of \( x \), making the inverse function \( f^{-1}(x) = \frac{x}{2} \).This shows how symbolic representation facilitates the process of finding inverse functions. It allows us to keep track of variables and operations clearly, transforming complex algebraic manipulations into a set of understandable steps.

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations and finding inverse functions. It involves rearranging equations to solve for a particular variable, often using basic operations like addition, subtraction, multiplication, and division.In the context of finding an inverse function, algebraic manipulation is what allows us to switch between \( f(x) \) and \( f^{-1}(x) \). For instance, with the function \( f(x) = 2x \), you find the inverse by first switching the equation to \( x = 2y \) and then solving for \( y \). By dividing both sides of the equation by 2, we isolate \( y \), resulting in \( y = \frac{x}{2} \). This approach is simple but powerful:

• Switch variables to focus on the desired unknown.
• Use inverse operations to isolate that variable.

Understanding these manipulations makes it easier to approach and solve problems involving inverse relationships in linear functions and beyond."