Problem 31 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=2 x+1\)

Step-by-Step Solution

Answer

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

Step 1: Set Up the Equation

To find the inverse function of \( f(x) = 2x + 1 \), start by replacing \( f(x) \) with \( y \). So, we write the equation as \( y = 2x + 1 \).

Step 2: Swap Variables

Swap the variables \( x \) and \( y \) in the equation. This is a standard approach for finding an inverse function: \( x = 2y + 1 \).

Step 3: Solve for y

Rearrange the equation \( x = 2y + 1 \) to solve for \( y \). Start by subtracting 1 from both sides: \( x - 1 = 2y \).

Step 4: Isolate y

Divide both sides of the equation \( x - 1 = 2y \) by 2 to solve for \( y \): \( y = \frac{x - 1}{2} \).

Step 5: Write the Inverse Function

The inverse function of \( f \) is denoted as \( f^{-1}(x) \). Thus, \( f^{-1}(x) = \frac{x - 1}{2} \).

Key Concepts

Solving Equations

When working with functions, you may often encounter instances where you need to solve equations to find unknown values. This process, called 'solving equations,' involves manipulating an equation to isolate a variable on one side. It requires various techniques and skills, such as understanding and applying inverse operations.

In the example of finding the inverse of the function \( f(x) = 2x + 1 \), the initial task is setting it up as \( y = 2x + 1 \) and then swapping the variables, resulting in \( x = 2y + 1 \). This swap is crucial, as it sets up the equation in a form where you can solve for the new variable, \( y \), which eventually becomes the expression for the inverse function.

To isolate \( y \), the equation \( x = 2y + 1 \) is manipulated by subtracting 1 from both sides, resulting in \( x - 1 = 2y \). The final step involves dividing both sides by 2, giving \( y = \frac{x - 1}{2} \). By performing these operations step-by-step, you've successfully solved the equation for \( y \), helping to determine the inverse function.

Linear Functions

A linear function is a function that creates a straight line when graphed. These functions can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In our example of \( f(x) = 2x + 1 \), it is apparent that this is a linear function. The slope \( 2 \) indicates the steepness and direction of the line, meaning for every unit change in \( x \), \( f(x) \) changes by two units. The y-intercept \( 1 \) is where the line crosses the y-axis.

Understanding linear functions is crucial when determining their inverse, as their straightforward nature aids in easy manipulation of their equations. By rearranging the original equation and switching variables, you can find the inverse function \( f^{-1}(x) = \frac{x - 1}{2} \), which is also linear, showcasing a simple relationship between the inputs and outputs.

Function Notation

Function notation provides a structured way to denote mathematical functions and understand how they operate. A typical function is written as \( f(x) \), where \( f \) is the function's name and \( x \) is the input variable.

In the exercise, \( f(x) = 2x + 1 \) is the original function. Using function notation, we can denote the inverse function of \( f \) as \( f^{-1}(x) \). This is not an exponent; rather, it's a special notation indicating that \( f^{-1} \) is the inverse of \( f \).

Function notation is valuable when solving equations. It helps to clearly identify what transformation or operation we are performing. In this case, writing \( f^{-1}(x) = \frac{x - 1}{2} \) specifies that this new function will provide an inverse relationship compared to the original \( f(x) \), allowing for accurate tracking of the functions and their inputs and outputs.