Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=x+5\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = x - 5\).

Step 1: Replace Function Notation with y

Identify that \(f(x) = x + 5\) can be rewritten using \(y\) for simplicity in manipulation. So, it becomes \(y = x + 5\).

Step 2: Solve for x

Rearrange the equation \(y = x + 5\) to solve for \(x\). Subtract 5 from both sides, giving \(x = y - 5\).

Step 3: Swap x and y

To find the inverse, swap \(x\) and \(y\) in the equation to express \(x\) in terms of \(y\), resulting in \(y = x - 5\).

Step 4: Rewrite in Inverse Function Notation

Replace \(y\) with \(f^{-1}(x)\) to express the inverse function. So, \(f^{-1}(x) = x - 5\).

Key Concepts

Function Notation

Function notation is simply a way to represent and work with functions in mathematics. Instead of using generic terms like "the output" for a function, function notation gives us clarity and precision by using symbols like \( f(x) \).
This notation specifies that \( f \) is a function and \( x \) is the variable put into the function. The result is \( f(x) \), which is the value you get once \( x \) goes through the function's rules.
For example, in the given exercise, the function notation is \( f(x) = x + 5 \). Here, \( f \) denotes the function, and \( x + 5 \) tells us how to process the input.

• \( f(x) \) represents a function with input \( x \).
• The expression after the equals sign tells us the operation to be performed on \( x \).

Understanding function notation is key because it helps in both expressing and manipulating functions efficiently for various operations, such as identifying inverse functions.

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of the unknown variable that makes the equation true.
When we deal with inverse functions, we often need to solve the equation to put it in a form that expresses one variable in terms of another.
In our example, starting with \( y = x + 5 \), we solve for \( x \) by isolating it on one side of the equation.

• Subtract 5 from both sides to get \( x = y - 5 \).

This process of isolating the variable involves operations that "undo" the current operation given in the function, which helps find the inverse function.
Solving equations accurately is crucial, as it directly affects the correct identification of inverse functions and other algebraic manipulations down the line.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions or equations to achieve a desired form. In the context of finding inverse functions, algebraic manipulation allows you to rearrange the given function to switch input and output.
Taking the equation \( y = x + 5 \) and subtracting 5 from each side is an example of algebraic manipulation.

• Perform operation: Subtract 5 from both sides, leading to \( x = y - 5 \).
• Reverse roles of variables to find inverse: Swap \( x \) and \( y \) to get \( y = x - 5 \).

Finally, rewrite as the inverse function notation \( f^{-1}(x) = x - 5 \).
Mastering algebraic manipulation is vital for tackling problems involving rearranging terms, simplifying expressions, and solving equations efficiently.
It ensures that transformations result in correct and simplified equivalent expressions.