Problem 46 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Inverse Function

To find the inverse of a function \(f(x)\), we need to find a function \(f^{-1}(x)\) such that \(f(f^{-1}(x)) = x\) for all \(x\) in the domain of \(f^{-1}(x)\) and \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f(x)\).

Step 2: Swap Variables

Begin finding the inverse by swapping the variables in the function. Set \(y = f(x)\), which leads to the equation \(y = \frac{x-1}{2}\). Swap \(x\) and \(y\) to get \(x = \frac{y-1}{2}\).

Step 3: Solve for Original Variable

Now, solve this equation for \(y\). Multiply both sides by \(2\) to eliminate the fraction: \(2x = y - 1\).

Step 4: Isolate the Variable

Add \(1\) to both sides to solve for \(y\): \(y = 2x + 1\).

Step 5: Write the Inverse Function

Once you have isolated \(y\), you have found the expression for the inverse function: \(f^{-1}(x) = 2x + 1\).

Key Concepts

Function Notation

Function notation is a way to express functions in a concise and standard manner. For example, the notation \(f(x)\) is commonly used to denote a function named 'f' with 'x' as its input. This signifies that the function takes 'x' as an argument and maps it to a specific output.

Function notation is crucial because it provides a clear and efficient way to communicate mathematical ideas. In the given problem, \(f(x) = \frac{x-1}{2}\), we're dealing with a function 'f' that describes how 'x' is transformed. When solving for its inverse, as highlighted in the exercise, we're looking for \(f^{-1}(x)\), which will reverse the effect of 'f'.

• The standard notation helps us easily denote operations and transformations.
• Using \(f(x)\) and \(f^{-1}(x)\) highlights the relationship between a function and its inverse.
• This notation makes it easier to track variables and operations, especially when performing algebraic manipulations.

Understanding and using function notation correctly enables students to approach problems systematically and efficiently.

Inverse Operations

Inverse operations are fundamental to mathematics, allowing us to "undo" actions performed by a function. If a function modifies input 'x,' its inverse function will reverse this modification, returning the input to its original state.

In simpler terms, if \(f(x)\) takes 'x' to another value, \(f^{-1}(x)\) brings it back to where it started. The key property of inverse functions is that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

• Inverse functions allow us to reverse a series of operations.
• For \(f(x) = \frac{x-1}{2}\), the inverse \(f^{-1}(x) = 2x + 1\) brings 'x' back by reversing the operations step-by-step.
• Understanding this concept helps in both algebraic and real-world problem solving.

Recognizing inverse operations in math allows us to think critically about how functions manipulate inputs and how we can use that to solve equations.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to simplify or solve equations. When finding an inverse function, algebraic manipulation is crucial in solving equations for a specific variable. In our example, we started with \(y = \frac{x-1}{2}\) and used algebraic steps to find its inverse.

To find \(f^{-1}(x)\):

• We first swapped variables, writing \(x = \frac{y-1}{2}\).
• Then, using multiplication, we cleared the fraction by multiplying both sides by \(2\), obtaining \(2x = y - 1\).
• Finally, by adding 1 to both sides, we solved for 'y', giving us \(y = 2x + 1\).

These steps show how algebraic manipulation transforms and isolates variables to solve equations efficiently. Understanding these techniques is essential for students, as it provides a foundation for solving a wide range of mathematical problems.