Problem 20 - College Algebra Essentials

Question

The functions in Exercises \(11-28\) are all one-to-one. For each function, a. Find an equation for \(f^{-1}(x),\) the inverse function. b. Verify that your equation is correct by showing that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\) $$f(x)=(x-1)^{3}$$

Step-by-Step Solution

Answer

The inverse function for the given function is \(f^{-1}(x) = \sqrt[3]{x} + 1\). By substitution, we have also verified that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), indicating that the found function is indeed the inverse.

Step 1: Find the Inverse Function

First, replace the function notation \(f(x)\) with \(y\). This gives us: \(y = (x-1)^{3}\). \nThen, switch the roles of \(x\) and \(y\) in order to find the inverse function: \(x = (y-1) ^{3}\). \nFinally, solve this equation for \(y\). The cube root of both sides gives us the inverse function: \(f^{-1}(x) = \sqrt[3]{x} + 1\).

Step 2: Verify the Inverse by Substitution

To verify that the found function is indeed the inverse, substitute it into the original function and the original function into it and simplify each expression. \nSubstitute \(f^{-1}(x)\) into \(f(x)\): \(f(f^{-1}(x)) = ((\sqrt[3]{x} + 1) - 1) ^{3}\) simplifies to \(f(f^{-1}(x)) = x\), which is what we expect. \nSimilarly, substitute \(f(x)\) into \(f^{-1}(x)\): \(f^{-1}(f(x)) = \sqrt[3]{((x - 1) ^{3})} + 1\) simplifies to \(f^{-1}(f(x)) = x\), as required for a valid inverse function.

Key Concepts

One-to-One Functions

Understanding one-to-one functions is key to mastering inverse functions. A function is considered "one-to-one" if every value of the function corresponds to exactly one value of the argument. In simpler terms, each output is linked to only one input.
Since each output is unique, one-to-one functions have exactly one inverse function. This is the defining characteristic of these functions.

• Imagine a series of different keys and locks where each key fits only one lock. Each key corresponds to one specific lock, which illustrates the one-to-one relationship.
• Mathematically, this means that if \( f(a) = f(b) \), then \( a = b \).
• Graphically, a function is one-to-one if any horizontal line intersects its graph at most once.

Identifying whether a function is one-to-one helps to determine if it is possible to find an inverse function.

Algebraic Equations

To find an inverse function, you often need to deal with algebraic equations. Start by replacing the function notation with a more manageable label, like \( y \), and then swap the roles of \( x \) and \( y \).
This involves solving the resulting equation for \( y \) to get the inverse function. For example, if you start with \( y = (x-1)^3 \), swap \( x \) and \( y \) to get \( x = (y-1)^3 \).
Solve for \( y \) by taking the cube root of both sides, resulting in the inverse function \( f^{-1}(x) = \sqrt[3]{x} + 1 \).

• Swapping the variables is key to finding the equation of the inverse.
• Reverse the operations used in the original function to solve for the inverse.
• Ensure your steps allow reversing the process, maintaining the balance of the equation.

Mastering these algebraic transformations is essential for uncovering inverse functions.

Function Verification

Verification of an inverse function involves confirming that the inverse successfully reverses the action of the original function. This is crucial to ensure the correctness of the derived inverse.
There are two main confirmations needed:

• Check that substituting \( f^{-1}(x) \) into \( f(x) \) returns \( x \): \( f(f^{-1}(x)) = x \).
• Verify that substituting \( f(x) \) into \( f^{-1}(x) \) also returns \( x \): \( f^{-1}(f(x)) = x \).

For instance, with \( f(x) = (x-1)^3 \) and \( f^{-1}(x) = \sqrt[3]{x} + 1 \), both substitutions should simplify back to \( x \).

• This verifies that the inverse function has been correctly calculated.
• Both conditions being met confirms the true inverse function, maintaining symmetry between the functions.
• Think of it as ensuring every output can be traced back to its unique input through the inverse.

Understanding and performing function verification solidifies the concept of inverse functions, offering the reliability needed in mathematical operations.