Problem 9 - College Algebra

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: Understand the Function

The given function is \(f(x) = 2 - x\). We know that an inverse function \(f^{-1}(x)\) is a function that, when composed with \(f(x)\), yields the original input value, \(x\). This means \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Step 2: Set Up the Equation for Inversion

To find the inverse, start by replacing \(f(x)\) with \(y\) to clarify the process and solve for \(x\). This gives us the equation \(y = 2 - x\).

Step 3: Solve for x

Re-arrange the equation \(y = 2 - x\) to solve for \(x\). This can be written as \(x = 2 - y\).

Step 4: Express the Inverse Function

Since \(x = 2 - y\), replace \(y\) with \(x\) to get the inverse function. Therefore, \(f^{-1}(x) = 2 - x\).

Key Concepts

Function Composition

Function composition involves combining two functions to form a single function. When we say we are composing functions, we apply one function to the results of another function. In simpler terms, think of it as plugging one function inside another.
For example, if you have two functions, say, \(f(x)\) and \(g(x)\), the composition of these functions, denoted as \(f(g(x))\), means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\).

• When dealing with function inversions, function composition helps verify that two functions are indeed inverses of each other.
• If \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), this confirms that \(f\) and \(f^{-1}\) are true inverses.

Understanding function composition is handy when working towards finding an inverse function since it is central to validating the correctness of the inversion.

Solving Equations

Solving equations is the process of finding the value of the unknown variable that makes the equation true. In the context of finding an inverse function, solving equations is a crucial step.
To find the inverse of a function, such as \(f(x) = 2 - x\), you set up an equation that represents the function in terms of \(y = 2 - x\).
Let's break it down:

• Initially, replace \(f(x)\) with \(y\) to assist in isolating the variable \(x\).
• Next, rearrange this equation to solve for \(x\), giving you \(x = 2 - y\).

Solving equations in this way allows us to unravel the operations that were applied to \(x\) in the original function and thus determine the inverse operations necessary for the inverse function.

Function Inversion

Function inversion is the process of finding a function \(f^{-1}(x)\) such that when it is composed with the original function \(f(x)\), it yields the identity value \(x\). In other words, the inverse function "undoes" what the original function does.
To find the inverse, start by setting the original function \(f(x)\) equal to \(y\). For the function \(f(x) = 2 - x\), this step looks like \(y = 2 - x\).
Next, solve this equation for \(x\). From \(y = 2 - x\), rearrange to find \(x = 2 - y\).
Finally, replace \(y\) with \(x\) to express the inverse function: \(f^{-1}(x) = 2 - x\).
This process ensures that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), confirming the functions are true inverses.

• Remember, not all functions have inverses. To have an inverse, a function must be one-to-one, meaning it passes both the horizontal and vertical line tests.

By understanding function inversion, we reveal the steps needed to reverse the effects of a given function, providing a comprehensive approach to working with inverse functions.