Problem 10 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

We are given the function \(f(x) = 3 - x\). This is a linear function where the output is subtracted from 3. The goal is to find the inverse function \(f^{-1}(x)\).

Step 2: Set the Function Equal to y

Replace \(f(x)\) with \(y\) to set up the equation as \(y = 3 - x\).

Step 3: Solve for x in Terms of y

To find the inverse, solve the equation \(y = 3 - x\) for \(x\). Start by isolating \(x\). Add \(x\) to both sides: \(y + x = 3\). Then subtract \(y\) from both sides: \(x = 3 - y\).

Step 4: Rewrite as the Inverse Function

Now that we have \(x = 3 - y\), swap \(x\) and \(y\) to write the inverse function: \(y = 3 - x\). So, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = 3 - x\).

Key Concepts

Linear Functions

Linear functions are some of the simplest and most foundational types of functions you'll encounter in mathematics. They are described by the equation of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions create straight lines when graphed, hence the name "linear". In our exercise, the function \(f(x) = 3 - x\) fits this category, even though it might look a bit different at first sight.
Let's break down \(f(x) = 3 - x\):

• Here, the slope \(m\) is \(-1\) because the coefficient of \(x\) is \(-1\).
• The y-intercept \(b\) is \(3\), meaning that the line crosses the y-axis at the point \( (0, 3) \).

This function tells us that for every unit increase in \(x\), the value of \(f(x)\) decreases by one. It is a simple way to model processes that have a constant rate of change, a common situation in the real world, like subtracting a fixed amount repeatedly.

Solving Equations

Solving equations is one of the most fundamental skills required in mathematics, and it involves finding the unknown value that makes the equation true. For our problem at hand, we started with the equation \(y = 3 - x\).
To solve for \(x\) in terms of \(y\), it’s all about isolating \(x\). Follow these steps:

• First, you add \(x\) to both sides to undo the subtraction of \(x\): \(y + x = 3\).
• Next, subtract \(y\) from both sides to solve for \(x\): \(x = 3 - y\).

By performing these algebraic operations, you rearrange the equation to express \(x\) in terms of \(y\). This method of isolating the variable is crucial for solving many kinds of equations, not just linear ones. Always remember the goal: get the variable by itself on one side of the equation.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. When working with inverse functions, we are particularly interested in compositions like \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). These compositions should return \(x\), because an inverse function essentially "undoes" the action of the function it's the inverse of.
Let's look closer at our exercise:

• The original function \(f(x) = 3 - x\) and its inverse \(f^{-1}(x) = 3 - x\) are identical in this case.
• When you compose them, you expect:
  • \(f(f^{-1}(x)) = f(3 - x) = 3 - (3 - x) = x\)
  • \(f^{-1}(f(x)) = f^{-1}(3 - x) = 3 - (3 - x) = x\)

This property of inverse functions is extremely useful in mathematics because it confirms that the original function and its inverse are correctly paired. They balance each other, ensuring that you can move back and forth between the transformed and original states.