Problem 6 - Intermediate Algebra

Question

To find the inverse of the function \(f(x)=2 x-3,\) we begin by replacing \(f(x)\) with \(y,\) and then we _________ x and \(y\)

Step-by-Step Solution

Answer

Swap x and y.

Step 1: Replace Function Notation with y

Start by writing the function as an equation with y: \[ y = 2x - 3 \].

Step 2: Swap x and y

Swap the positions of x and y in the equation, turning it into:\[ x = 2y - 3 \].

Step 3: Solve for y

Rearrange the equation to solve for y. First, add 3 to both sides to get:\[ x + 3 = 2y \]. Then, divide both sides by 2 to isolate y:\[ y = \frac{x + 3}{2} \].

Step 4: Write the Inverse Function

Since \( y = \frac{x + 3}{2} \) represents the inverse of the original function, write it in inverse function notation:\[ f^{-1}(x) = \frac{x + 3}{2} \].

Key Concepts

Understanding Function Notation

Function notation is a way to name a function using the symbol \( f(x) \), which represents the output of the function \( f \) for the input \( x \). It simplifies expressing how each input corresponds to an output. For example, in the function \( f(x) = 2x - 3 \), \( f(x) \) tells us the result of doubling \( x \) and then subtracting 3.
It's essential to view \( f(x) \) not just as a label but as a formula showing the relationship between the input \( x \) and its output. Understanding this helps in easily manipulating expressions, especially when finding an inverse.
When finding inverses, we often start by replacing \( f(x) \) with \( y \) for simplicity, turning the function into an equation \( y = 2x - 3 \). Doing so sets the stage for finding the inverse by allowing us to manipulate the equation freely without the function notation distracting us.

Steps for Solving Equations

Solving equations involves finding the unknown values that make the equation true. When swapping \( x \) and \( y \) in the equation \( x = 2y - 3 \), the goal becomes isolating \( y \).
Follow these steps to solve the equation:

• Add 3 to both sides: \( x + 3 = 2y \).
• Divide both sides by 2 to solve for \( y \) and isolate it: \( y = \frac{x + 3}{2} \).

Each step systematically narrows down to express \( y \) clearly in terms of \( x \).
These algebraic manipulations are crucial to reveal the relationships between variables and to understand how the function behaves in reverse. Practicing solving equations helps you become efficient in converting problems into simpler forms, making them easier to tackle.

Inverse Operation and Its Importance

The idea of an inverse operation is foundational in mathematics. It's essentially doing operations in reverse to retrieve an original value or state. Inverse functions work similarly: they "undo" the operation of the original function.
For the function \( f(x) = 2x - 3 \), its inverse \( f^{-1}(x) = \frac{x + 3}{2} \) helps find the original input before the function was applied. This is like solving a mystery in reverse, where each step back gives insight into the function's original settings.

• Start with finding an inverse by swapping \( x \) and \( y \).
• Then solve for \( y \) to reveal the inverse equation.
• Now, this inverse can find pre-images of any output from the original function.

Understanding inverse operations builds confidence and skills in transforming equations and recognizing underlying patterns, crucial for tackling complex problems in calculus and beyond.