Problem 7 - College Algebra

Question

For the following exercises, find \(f^{-1}(x)\) for each function. $$ f(x)=x+3 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

Given the function is \( f(x) = x + 3 \). This is a linear function of the form \( f(x) = x + c \), where \( c = 3 \). We need to find the inverse function \( f^{-1}(x) \).

Step 2: Replace f(x) with y

Set \( y = f(x) \), so we have \( y = x + 3 \). This helps in making the variable clearly recognizable for manipulation.

Step 3: Solve for x

Rearrange the equation to solve for \( x \). Start with \( y = x + 3 \). Subtract 3 from both sides to get \( x = y - 3 \).

Step 4: Switch the Variables

Now that we have \( x = y - 3 \), switch \( x \) and \( y \) to find the inverse: \( y = x - 3 \).

Step 5: Write the Inverse Function

Thus, the inverse function is \( f^{-1}(x) = x - 3 \).

Key Concepts

Linear Function

A linear function is perhaps one of the simplest types of functions in mathematics. It is expressed in the form \( f(x) = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept. In our case, the function is \( f(x) = x + 3 \), which tells us two things: the slope \( a = 1 \), indicating a 45-degree angle line if plotted, and the y-intercept \( b = 3 \), denoting that the line crosses the y-axis at (0, 3). Linear functions graph as straight lines, making them predictable and easy to understand.

• Slope: Determines the angle of the line.
• y-intercept: Where the graph crosses the y-axis.

Recognizing these properties helps in visualizing and manipulating linear functions effectively.

Solving Equations

When dealing with functions, especially when finding inverses, solving equations becomes a critical skill. To solve an equation means to isolate the variable of interest, usually \( x \), on one side of the equation.In the example given, we start with \( y = x + 3 \). To solve for \( x \), we need to undo the operation performed on it.

• Subtract 3 from both sides: \( x = y - 3 \).

Keeping operations balanced is key. Whatever you do to one side, do it to the other. This ensures the equality still holds, allowing you to isolate the variable and solve the equation correctly.Solving these kinds of equations is foundational, allowing further manipulations required to find inverse functions or other function properties.

Function Notation

Function notation is a precise way of denoting functions and their operations. It uses \( f(x) \) to represent the function values, making it easier to plug in different inputs and analyze outputs. In \( f(x) = x + 3 \), \( f \) is the function name, and \( x \) is the variable.Switching to \( y \) helps to break things down. Hence, we write, \( y = x + 3 \), giving us the same information but in a manipulable format.

• {{ f(x) }}: This shows what the function returns when you input \( x \).
• {{ y = f(x) }}: Helps in transforming the function into a solvable equation.

Understanding function notation is crucial because it's ubiquitous in mathematics, making it easier to communicate complex ideas succinctly.

Inverse Operations

Inverse operations reverse the effect of the original operation. They are fundamental in finding inverse functions. For example, if the function operation adds 3, its inverse operation should subtract 3.In the function \( f(x) = x + 3 \), to find its inverse \( f^{-1}(x) \), we need to perform the opposite operation:

• Subtract 3: This reverses the addition, helping isolate the original input \( x \).
• Switch the variables to complete finding the inverse: After solving for \( x \), swapping \( x \) and \( y \) positions gives the inverse form \( y = x - 3 \).

The inverse function \( f^{-1}(x) = x - 3 \) now effectively undoes what \( f(x) = x + 3 \) does, providing a full circle understanding of how functions and their inverses operate. Recognizing and applying inverse operations is vital for reversing processes within mathematical problems.