Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=2-x\)

Step-by-Step Solution

Answer

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

Step 1: Replace f(x) with y

Start with the given function: \( f(x) = 2 - x \). Replace \( f(x) \) with \( y \) to set up the equation: \( y = 2 - x \). This substitution will help us find the inverse function by solving for \( x \) in terms of \( y \).

Step 2: Solve for x in terms of y

We need to express \( x \) in terms of \( y \). Start with the equation \( y = 2 - x \). Add \( x \) to both sides to get \( x + y = 2 \). Then, subtract \( y \) from both sides to solve for \( x \): \( x = 2 - y \).

Step 3: Replace x with f^{-1}(x) and y with x

To express the inverse function, swap \( x \) and \( y \), changing \( y \) to \( x \) and \( x \) to \( f^{-1}(x) \). This gives us \( f^{-1}(x) = 2 - x \).

Key Concepts

Function Notation

When dealing with functions, it is common to encounter different ways that functions are represented, known as function notation. The notation \( f(x) \) indicates that \( f \) is a function of \( x \), the input variable. In simple terms, it is like a formula or a machine where you put in a value for \( x \) and get out a result.

• The expression \( f(x) = 2 - x \) tells us that for any input \( x \), to find the output, we subtract \( x \) from 2.
• Inverse functions are often represented with \( f^{-1}(x) \) notation. This means that this new function undoes the work of the original function.

This notation helps us clearly see the relationship between inputs and outputs and also allows us to communicate the idea of reversing a function with the inverse.

Solving Equations

Solving equations is a crucial step in finding inverse functions, as you need to rearrange the function equation to express the input variable in terms of the output variable. For the function \( f(x) = 2 - x \), we start by letting \( y = f(x) \), which translates to \( y = 2 - x \). Our goal is to manipulate this equation to solve for \( x \) as a function of \( y \).

• First, to isolate \( x \), we add \( x \) to both sides: \( x + y = 2 \).
• Then, subtract \( y \) from both sides, which gives us \( x = 2 - y \).

Once you solve for \( x \), you can appropriately swap \( x \) and \( y \) to express the inverse function in standard form.

Inverse Operations

Inverse operations are key in finding the inverse of a function. The concept of inverse operations is about using operations that reverse or "undo" each other. In the original function \( f(x) = 2 - x \), we subtract \( x \) from 2 to find the output.

• The inverse function, \( f^{-1}(x) \), must undo this subtraction by using addition, retracing steps to the original input.
• When we set up \( y = 2 - x \) and solve for \( x \), we reverse the operations to express \( x \) in terms of \( y \), leading to \( x = 2 - y \).

Finally, to find the inverse function, we switch the variables, yielding \( f^{-1}(x) = 2 - x \), reinforcing how inverse operations effectively reverse the function's process.