Problem 15 - Elementary and Intermediate Algebra

Question

Complete each solution. Find the inverse of \(f(x)=2 x-3\) (SOLUTION CAN'T COPY)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{x + 3}{2}\).

Step 1: Substitute y for f(x)

To find the inverse of a function, we first replace the function notation \(f(x)\) with \(y\). So, we begin with \(y = 2x - 3\). This change of notation helps simplify the algebraic manipulations that follow.

Step 2: Swap x and y

The next step involves switching the roles of \(x\) and \(y\). Replace \(y\) with \(x\) and \(x\) with \(y\) to get the equation \(x = 2y - 3\). This step is crucial as the inverse function essentially swaps the input and output values.

Step 3: Solve for y

Now, our goal is to solve for \(y\) in terms of \(x\). Start by adding 3 to both sides of the equation:\[ x + 3 = 2y \].Then, divide both sides by 2 to isolate \(y\):\[ y = \frac{x + 3}{2} \].

Step 4: Write the Inverse Function

Now that we have \(y\) in terms of \(x\), we can write the inverse function. Replace \(y\) with \(f^{-1}(x)\) to get the inverse function:\[ f^{-1}(x) = \frac{x + 3}{2} \].

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in mathematics that involves rearranging equations and expressions to solve for unknown variables. In the exercise given, finding the inverse of the function required us to perform several algebraic manipulations.

• Firstly, we began with the equation obtained by rewriting the function notation: \( y = 2x - 3 \).
• We then swapped \( x \) and \( y \) to represent the inverse: \( x = 2y - 3 \).
• The goal was to solve for \( y \), requiring us to perform algebraic steps like adding 3 on both sides: \( x + 3 = 2y \), and then dividing by 2: \( y = \frac{x + 3}{2} \).

These manipulations are essential to isolate the variable and derive the function that represents the inverse relationship. Mastery of algebraic manipulation allows for solving equations efficiently and accurately.

Function Notation

Function notation provides an effective way to represent mathematical functions, showcasing the input-output relationship clearly. In this exercise, we initially had the function \( f(x) = 2x - 3 \).Function notation helps in:

• Describing the function's operation (e.g., multiplying by 2, subtracting 3).
• Indicating the variable used (\( x \)) and how it gets processed to yield a result.

When finding the inverse, we use function notation once more to describe the inverse function as \( f^{-1}(x) \). This indicates that the input-output roles of the original function have been reversed. Thus, function notation plays a vital role not just in expressing functions but also in understanding their inverses, broadening the scope of function analysis.

Solving Equations

Solving equations is a critical step in finding inverse functions, requiring careful operations to isolate variables. Throughout the process, every operation we apply to one side of the equation must be done to the other. This maintains the balance and equality of the equation.With the equation \( x = 2y - 3 \):

• We started by adding 3 to both sides to counter the subtraction: \( x + 3 = 2y \).
• Next, division by 2 removed the multiplication attached to \( y \), leading to \( y = \frac{x + 3}{2} \).

By these steps, we ensure \( y \) is isolated correctly, giving us the solution to write the inverse function. Solving equations involves understanding operations and properties of equality, a foundational skill in algebra.