Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. 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

Step 1: Replace f(x) with y

In the given function \( f(x) = 2x - 3 \), start by replacing \( f(x) \) with \( y \). This gives us the equation: \( y = 2x - 3 \).

Step 2: Swap x and y

The process to find the inverse of a function involves swapping the variables \( x \) and \( y \). This gives us the equation: \( x = 2y - 3 \).

Key Concepts

Function Notation

Function notation is a way to name a function using symbols. You'll often see it represented as \( f(x) \) for functions. This notation uses the letter \( f \) and a variable in parentheses to show how a particular variable, like \( x \), is changed by the function.

For example, in the function \( f(x) = 2x - 3 \), \( f(x) \) is the name of the function. The expression \( 2x - 3 \) tells you what to do with \( x \). You multiply \( x \) by 2, then subtract 3.

Using function notation makes it easy to keep track of multiple functions when you're working through math problems. It also helps to identify the input and output clearly. If \( x \) is 5, then \( f(x) \) becomes \( f(5) \), which shows how \( x \) is specifically processed in the function.

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. When solving, we perform operations to isolate the variable on one side of the equation.

For inverses, solving equations helps us to express one variable in terms of another. Take the example \( y = 2x - 3 \). To isolate \( x \), you would reverse the operations:

• Add 3 to both sides: \( y + 3 = 2x \).
• Then divide by 2: \( \frac{y+3}{2} = x \).

This shows that solving the equation involves systematically reversing operations until the variable stands alone. The resulting equation gives us insights into how the original function changes.

Switching Variables

Switching variables is an essential step in finding the inverse of a function. It involves exchanging the roles of the independent and dependent variables in an equation.

In the function \( f(x) = 2x - 3 \), we start by replacing \( f(x) \) with \( y \), leading to the equation \( y = 2x - 3 \).

To find the inverse, switch \( x \) with \( y \), resulting in \( x = 2y - 3 \). This switching sets the stage for solving the new equation for \( y \).

Why switch variables? Because inverses are about reversing the original process. They let you find the original input if you know the output. This concept is a cornerstone of understanding and working with inverse functions.