Problem 32 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=6-x\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = 6 - x\).

Step 1: Understand the Definition of an Inverse Function

An inverse function reverses the operation of the original function. If a function is represented by \(y = f(x)\), then its inverse, represented by \(x = f^{-1}(y)\), will swap the roles of \(x\) and \(y\). For \(f(x) = 6 - x\), we are looking for a function \(f^{-1}(x)\) such that \(f(f^{-1}(x)) = x\).

Step 2: Replace \(f(x)\) with \(y\)

Instead of \(f(x) = 6 - x\), rewrite it as \(y = 6 - x\). This makes it easier to solve for \(x\) in terms of \(y\), which is necessary for finding the inverse.

Step 3: Solve for \(x\)

Starting with the equation \(y = 6 - x\), isolate \(x\) by performing the following operations:1. Add \(x\) to both sides to get: \(x + y = 6\).2. Subtract \(y\) from both sides to solve for \(x\): \(x = 6 - y\).

Step 4: Swap the Variables

To express the inverse function, swap the roles of \(x\) and \(y\). This makes the equation \(y = 6 - x\), where \(y\) can now be considered as \(f^{-1}(x)\).

Step 5: Write the Inverse Function

The inverse function of the original function \(f(x) = 6 - x\) is given by: \(f^{-1}(x) = 6 - x\). In this case, the function is equal to its inverse.

Key Concepts

Function Operations

Function operations are essential tools that allow us to combine, manipulate, and understand functions in various ways. One of the fundamental operations is finding the inverse of a function. Inverse functions 'undo' what the original function does. The step-by-step process of finding an inverse involves a series of operations meant to swap the input and output values. Let's break down how function operations facilitate this process.

When dealing with function operations, it's crucial to understand how to manage expressions. For example, if we want to find the inverse of a function like \( f(x) = 6 - x \), we perform operations like addition and subtraction to manipulate the expression. These operations help us express \( x \) in terms of \( y \), or vice versa. It's a step-by-step task:

• Rewrite the equation by replacing \( f(x) \) with \( y \).
• Isolate \( x \) to express it in terms of \( y \).
• Once isolated, swap the variables to finalize the inverse function form.

These basic function operations are directly tied to solving equations, which is another critical concept in understanding inverse functions.

Solving Equations

Solving equations is a skill that underpins finding inverse functions. To solve an equation, one must manipulate it to isolate a particular variable. This typically involves using inverse operations such as addition, subtraction, multiplication, or division.

In our original problem, we are tasked with finding the inverse of \( f(x) = 6 - x \). Here, solving the equation involves rearranging terms to make \( x \) the subject. The steps include:

• Starting with \( y = 6 - x \).
• Rearranging to get \( x + y = 6 \).
• Isolating \( x \) yields \( x = 6 - y \).

This isolation is possible through skillfully applying function operations as mentioned earlier. Once the variable is isolated, we can proceed to swap values to determine the inverse. Each of these steps is crucial for understanding how variables interrelate within the equation.

Solving such equations requires patience and practice, but it is a foundational aspect of working with functions and their inverses.

Functions and Their Properties

Functions are the backbone of algebra and calculus, and understanding their properties is critical for any mathematical journey. One intriguing property of certain functions, like \( f(x) = 6 - x \), is that they are self-inverse. This means their inverse function is identical to the original function. This property stems from the way the operations within the function are balanced.

To explore a function's properties, you need to consider:

• What type of operations it involves (like addition or subtraction).
• How those operations affect the input and output.
• The symmetry or balance of the function, which can reveal self-inverse characteristics.

In the specific case of \( f(x) = 6 - x \), each input transformation applies equally in the reverse when finding the inverse, hence it remains unchanged. This property simplifies the process of finding the inverse function and serves as a valuable insight into the function's behavior.

By scrutinizing the operations and transformations, students can grasp how inverse functions work and appreciate the elegance of self-inverse functions.