Problem 43 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=-2 x+10 $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = -\frac{x}{2} + 5\).

Step 1: Understanding the Function

We start with the function given as \(f(x) = -2x + 10\). Our goal is to find its inverse, \(f^{-1}(x)\). This means we want to find a function that, when composed with \(f(x)\), will return the input value, \(x\). Simply put, if \(y = f(x)\), then \(f^{-1}(y) = x\).

Step 2: Switching Variables

To find the inverse function, we switch the roles of \(x\) and \(y\). So, we rewrite the function \(y = -2x + 10\) as \(x = -2y + 10\).

Step 3: Solving for y

Now, solve the equation \(x = -2y + 10\) for \(y\). Start by isolating \(y\). 1. Subtract 10 from both sides: \(x - 10 = -2y\) 2. Divide every term by \(-2\): \(y = \frac{x - 10}{-2}\) This simplifies to: \(y = -\frac{x}{2} + 5\)

Step 4: Expressing the Inverse Function

Now that \(y = -\frac{x}{2} + 5\), we restate this in function notation as the inverse function: \(f^{-1}(x) = -\frac{x}{2} + 5\).

Key Concepts

Function Composition

Function composition involves combining two functions in a specific order. This concept is essential in understanding inverse functions. For a function \(f(x)\) to have an inverse, i.e., \(f^{-1}(x)\), the composition \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) should return the original input \(x\).

• \(f(f^{-1}(x)) = x\): This means applying the inverse function first, then the original function gets us back to our starting point \(x\).
• \(f^{-1}(f(x)) = x\): This means starting with \(f(x)\) and then applying the inverse also gets back to the input \(x\).

When dealing with linear functions like \(f(x) = -2x + 10\), verifying the correct expression for \(f^{-1}(x) = -\frac{x}{2} + 5\) involves checking these compositions. If they simplify to \(x\), the inverse is accurate.

Solving Equations

Solving equations is a fundamental skill in finding inverse functions. After switching the variables in a function, you will need to solve for the new output variable to identify the inverse function.
In the given exercise, we swap \(x\) and \(y\) in the function \(y = -2x + 10\), resulting in \(x = -2y + 10\). Now, solving for \(y\) looks like this:

• Subtract 10 from both sides: \(x - 10 = -2y\).
• Then divide by \(-2\) to isolate \(y\): \(y = \frac{x - 10}{-2}\).
• This simplifies to the expression: \(y = -\frac{x}{2} + 5\).

This process of solving for \(y\) after switching variables is the essence of discovering the inverse function. Always remember that solving equations is about maintaining balance and transforming the function to reveal the inverse.

Algebraic Manipulation

Algebraic manipulation is crucial for deriving inverse functions because it allows us to rearrange and solve equations efficiently. In our example, we begin with the equation \(x = -2y + 10\) and use simple algebraic techniques to isolate \(y\).
The main tools of algebraic manipulation involved in this exercise include:

• Subtraction: Adjust the equation by moving terms across the equals sign, like subtracting 10 from both sides to maintain equation balance: \(x - 10 = -2y\).
• Division: Once subtraction is done, we divide by the coefficient of \(y\), which is \(-2\), to solve for \(y\): \(y = -\frac{x}{2} + 5\).

These techniques make algebraic manipulation critical, especially when finding the inverse, as they allow for the clear and precise expression of the functions. Mastery of simple manipulation forms the backbone of more complex algebraic operations.