Problem 7 - College Algebra with Corequisite Support

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: Understand the Problem

We are given the function \(f(x) = x + 3\) and we need to find its inverse, \(f^{-1}(x)\). The inverse function should reverse the effect of the original function.

Step 2: Replace Function Notation with y

Begin by setting \(y = f(x) = x + 3\). Our aim is to express \(x\) in terms of \(y\) to find the inverse function.

Step 3: Swap Variables

Swap the roles of \(x\) and \(y\) in the equation. This gives us \(x = y + 3\).

Step 4: Solve for y

Solve the equation from the previous step for \(y\). Subtract 3 from both sides to get \(y = x - 3\).

Step 5: Write the Inverse Function

Now that we have \(y = x - 3\), we can write the inverse function as \(f^{-1}(x) = x - 3\).

Key Concepts

Function Notation

Function notation is a fundamental concept in mathematics that allows us to work with functions in a clear and organized way. When using function notation, we denote functions with a letter, such as \(f\), followed by \(x\) in parentheses: \(f(x)\). This tells us that \(x\) is the input variable and \(f(x)\) is the output, or value of the function at \(x\).

Function notation is important because it helps us to track which function we are working with, especially when dealing with multiple functions. It also helps in expressing complex operations on functions, such as finding their inverse.

• When using function notation, always remember that \(f(x)\) is not a multiplication of \(f\) and \(x\), but a representation of the function's output when \(x\) is the input.
• Inverse functions, like \(f^{-1}(x)\), are denoted to show that the operation reverses the effect of the original function.
• Function notation provides a useful shorthand in mathematics. It allows us to distinguish between different operations and transformations on variables.

Solving Equations

In mathematics, solving equations is a crucial skill that involves finding the values of variables that make an equation true. In the context of inverse functions, solving equations helps us derive the inverse by rearranging terms.

Here is a simple example with the given function \(f(x) = x + 3\):

• First, express \(f(x)\) as \(y\): \(y = x + 3\).
• The goal of finding an inverse is to express \(x\) in terms of \(y\), effectively "undoing" the function's operation.
• Swap \(x\) and \(y\) to represent reversing the function: \(x = y + 3\).
• Solving for \(y\), subtract 3 from both sides to isolate \(y\): \(y = x - 3\).

This step-by-step manipulation demonstrates how solving equations is used to find inverse functions.

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic techniques and principles to transform equations or expressions into a different form. This skill is especially needed when finding inverse functions to restructure the equation.

Using the example of the function \(f(x) = x + 3\), algebraic manipulation helps in transforming it to find the inverse:

• First, rewrite the function equation: \(y = x + 3\).
• Through algebraic manipulation, swap \(x\) and \(y\) to think about reversing the operations: \(x = y + 3\).
• Remove the constant on the right side by subtracting 3 from both sides: \(y = x - 3\).

These steps involve applying arithmetic operations, such as addition and subtraction, which are part of algebraic manipulation.

Understanding algebraic manipulation makes it easier to handle various mathematical operations and transformations. It ensures proper handling of expressions and equations, leading to correct solutions, such as determining the inverse of a function.