Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=(x-1) / 2$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = (x - 1) / 2\) is \(f^{-1}(x) = 2x + 1\). Substituting the inverse function into the original function and vice versa both yield \(x\), verifying that these functions are indeed inverses of each other.

Step 1: Finding the inverse function

To find the inverse of the function \(f(x)\), swap \(x\) and \(y\) and solve for \(y\):1. Replace \(f(x)\) with \(y\) to get \(y = (x - 1) / 2\).2. Swap \(x\) and \(y\) to get \(x = (y - 1) / 2\).3. Solve for \(y\), which yields \(y = 2x + 1\).The inverse of \(f(x)\), denoted as \(f^{-1}(x)\), is therefore \(f^{-1}(x) = 2x + 1\).

Step 2: Verifying \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into the function \(f\):1. Replace \(x\) in \(f(x)\) with \(f^{-1}(x)\) to get \(f(f^{-1}(x)) = ((2x + 1) - 1) / 2\).2. Simplify the expression to get \(f(f^{-1}(x)) = x\).This verifies the equation \(f(f^{-1}(x)) = x\).

Step 3: Verifying \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) into the function \(f^{-1}\):1. Replace \(x\) in \(f^{-1}(x)\) with \(f(x)\) to get \(f^{-1}(f(x)) = 2((x - 1) / 2) + 1\).2. Simplify the expression to get \(f^{-1}(f(x)) = x\).This verifies the equation \(f^{-1}(f(x)) = x\).

Key Concepts

Function Verification

Function verification is a crucial step in understanding the relationship between a function and its inverse. After deriving what you believe to be the inverse function, it’s important to ensure that processing an input through the original function and then through its inverse (and vice versa) will return the original input. This process involves two checks, often referred to as the **composition checks**, where:

• First, you substitute the inverse function into the original function, denoted as \( f(f^{-1}(x)) = x \).
• Then, you substitute the original function into the inverse function, written as \( f^{-1}(f(x)) = x \).

Each check should simplify back to the identity function of \( x \). If both these equations hold true, you have successfully verified that the functions are indeed inverses of each other. It's like a mathematical handshake, confirming the accuracy of your work!

Inverse of a Linear Function

Finding the inverse of a linear function involves several straightforward steps. For the function given as \( f(x) = \frac{x - 1}{2} \), the goal is to rearrange and solve it in terms of \( y \), and then swap \( x \) and \( y \) to solve for the new expression.

• Replace the function format with \( y \): \( y = \frac{x - 1}{2} \).
• Interchange \( x \) and \( y \) to begin finding the inverse: \( x = \frac{y - 1}{2} \).
• Solve this equation for \( y \), yielding \( y = 2x + 1 \).

Thus, the inverse function, \( f^{-1}(x) \), is determined to be \( 2x + 1 \). This transformation is typical for linear functions, where solving essentially involves algebraic manipulation to isolate the variable when interchanged. It's like flipping a procedure around, letting you reverse its effects.

Function Composition

Function composition is a fundamental concept where the output of one function becomes the input of another. This is particularly useful in function verification, ensuring both the function and its inverse are correctly calculated.

• Substituting the inverse into the original function \( f(f^{-1}(x)) \), confirms that upon simplification, it should return \( x \).
• Similarly, substituting the original function into the inverse function \( f^{-1}(f(x)) \) should also result in \( x \).

These operations are analogous to testing whether two functions "undo" each other properly. If the compositions return the variable from the beginning, this indicates that the operations are reversible, affirming that the functions are valid inverses. It’s like ensuring both pathways through a maze lead you back to the start, demonstrating their interconnectedness and reliability.