Problem 16 - Precalculus Essentials

Question

The functions 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 $$f\left(f^{-1}(x)\right)=x \text { and } f^{-1}(f(x))=x$$ $$f(x)=3 x-1$$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = 3x - 1\) is \(f^{-1}(x) = (x + 1) / 3\), and it has been verified to be correct.

Step 1: Finding the Inverse

To find the inverse, we first write the function \(f(x) = 3x - 1\) as \(y = 3x - 1\). Then, we exchange the roles of \(x\) and \(y\), getting \(x = 3y - 1\). After this, solving the equation for \(y\), we get \(y = (x + 1) / 3\). Hence, \(f^{-1}(x) = (x + 1) / 3\).

Step 2: Verifying the Inverse Part I

To verify that we have found the correct inverse, we plug \(f^{-1}(x)\) into \(f(x)\), substituting every \(x\) in \(f(x) = 3x - 1\) with \((x + 1) / 3\). This gives \(f(f^{-1}(x)) = 3((x + 1) / 3) - 1\). Simplifying, this equals \(x\), which is what we wanted to show for this step of the verification.

Step 3: Verifying the Inverse Part II

The second part of the verification is to substitute \(f(x)\) into \(f^{-1}(x)\), replacing all \(x\) in \((x + 1) / 3\) with \(3x - 1\). This gives us \(f^{-1}(f(x)) = ((3x - 1) + 1) / 3\). Simplifying, we get \(f^{-1}(f(x)) = x\), which completes the verification that \(f^{-1}(x)\) is indeed the inverse of \(f(x)\).

Key Concepts

One-to-One Function

A one-to-one function is a special type of function in mathematics. It is also called an injective function. For a function to be one-to-one, every element in the domain must map to a unique element in the range. This means that no two different inputs should have the same output.
In simpler terms:

• If you have two different inputs, you must get two different outputs.
• This ensures that the function is "reversible," making it possible to find an inverse.

For example, consider the function given by the exercise, \(f(x) = 3x - 1\). If you take two distinct values, say \(x_1\) and \(x_2\), and plug them both into \(f(x)\), the outputs will be different as long as \(x_1 eq x_2\). This quality helps when finding an inverse function because it guarantees the process of reversing the input and output pairs.

Function Verification

Verification of a function, especially when determining its inverse, is crucial to ensure accuracy in calculations. Once an inverse function is found, it needs to be checked to verify that it truly undoes the work of the original function.
Verification consists of demonstrating two main properties:

• Plugging the inverse function into the original function should return the starting point \(x\); that is, \(f(f^{-1}(x)) = x\).
• Plugging the original function into the inverse function should again return the initial value \(x\); thus, \(f^{-1}(f(x)) = x\).

Following these steps ensures that the inverse function was derived correctly. In the exercise solution, using \(f(x) = 3x - 1\) and \(f^{-1}(x) = (x + 1)/3\), both verifications derive back to \(x\), validating that \(f^{-1}(x)\) is the true inverse.

Solving Equations

Solving equations is a process that often involves finding a value for a variable that makes a given equation true. This involves several steps like isolating the variable or simplifying expressions.
In the case of finding an inverse function, solving equations is at the core. Here's how it works for our exercise:

• Rewrite the original function, \(f(x) = 3x - 1\), as \(y = 3x - 1\).
• Switch around the variables by exchanging \(x\) and \(y\), resulting in \(x = 3y - 1\).
• Isolate \(y\) by solving for it, giving you \(y = (x + 1)/3\).

This series of steps transforms \(x = 3y - 1\) into \(y = (x + 1)/3\), which represents our inverse function \(f^{-1}(x)\). Being comfortable with solving equations is vital for tasks like these as they are essential when rearranging and pinpointing the inverse correctly.

Inverse Equation Derivation

Deriving the inverse of a function involves reversing the processes and operations applied in the original function. It's like reconstructing a path in reverse, step by step, to find the way back.
To derive the inverse equation, follow these steps:

• Start with the function written as an equation, such as \(y = 3x - 1\).
• Swap the variables \(x\) and \(y\) to plan the "reverse"; here it's \(x = 3y - 1\).
• Isolate \(y\) to express it solely in terms of \(x\), leading to \(y = (x + 1)/3\).

This final equation \(y = (x + 1)/3\) becomes your \(f^{-1}(x)\) or the inverse function. This is derived systematically by reversing every operation of the original function \(f(x)\). Understanding this process helps to grasp how functions and their inverses relate and transform one another.