Problem 7 - Precalculus with Limits

Question

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1} (x)) = x\) and \(f^{-1} (f (x)) = x\). \(f(x) = 6x\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 6x\) is \(f^{-1}(x) = x/6\), and it is verified that \(f(f^{-1} (x)) = x\) and \(f^{-1} (f (x)) = x\).

Step 1: Swap \(x\) and \(f(x)\) in the equation

By swapping \(x\) and \(f(x)\), we get \(x = 6f^{-1}(x)\) where \(f^{-1}(x)\) represents the inverse of the function \(f(x)\).

Step 2: Solve the resulting equation for \(f^{-1}(x)\)

We can solve for \(f^{-1}(x)\) by dividing both sides of the equation \(x = 6f^{-1}(x)\) by 6, which gives us \(f^{-1}(x) = x/6\). This is the inverse function.

Step 3: Verify that \(f(f^{-1} (x)) = x\)

We substitute \(f^{-1}(x)\) into \(f(x)\) which gives us \(f(f^{-1}(x)) = 6*(x/6) = x\). So this equation holds true.

Step 4: Verify that \(f^{-1} (f (x)) = x\)

We substitute \(f(x)\) into \(f^{-1}(x)\) which gives us \(f^{-1}(f(x)) = (6x)/6 = x\). So this equation also holds true.

Key Concepts

Function Verification

Verifying if a function is indeed the inverse of another is an essential concept in mathematics. The process involves checking two main conditions:

• First, ensure that when the inverse function, denoted as \(f^{-1}(x)\), is applied to the original function \(f(x)\), we get back \(x\). This is stated as \(f(f^{-1}(x)) = x\).
• Secondly, when the original function is applied to the inverse function, it should also return \(x\), expressed as \(f^{-1}(f(x)) = x\).

These verifications confirm that the functions genuinely undo each other's actions. This is particularly crucial in fields like calculus and precalculus, where understanding function composition helps in finding simpler forms of complex expressions. In the context of our task, both verifications hold true for the function \(f(x) = 6x\) and its inverse \(f^{-1}(x) = \frac{x}{6}\). This shows they are truly inverses of one another.

Swapping Variables

The first step in finding an inverse function is to swap the variables in the function equation. This is because the inverse function essentially reverses the input and output roles.

• Start from \(f(x) = 6x\) and replace \(f(x)\) with \(y\), so you have \(y = 6x\).
• Then, swap \(x\) and \(y\) to reflect the inverse relationship, resulting in \(x = 6y\).

This swapping lays the groundwork for solving for \(y\), which represents \(f^{-1}(x)\). It allows you to then algebraically manipulate the equation to isolate and express the inverse function clearly.

Solving Equations

Once variables have been swapped in the function equation, the next step is to solve for the inverse function. This involves isolating \(f^{-1}(x)\). Let's break it down:

• Starting from the swapped equation \(x = 6y\), where \(y\) denotes \(f^{-1}(x)\), your goal is to solve for \(y\).
• Do this by dividing both sides of the equation by 6. This gives \(y = \frac{x}{6}\).

Thus, the inverse function is \(f^{-1}(x) = \frac{x}{6}\). This simple algebraic operation is crucial in finding inverses, ensuring that you reverse the original operation conducted by the function.

Precalculus

In precalculus, understanding functions and their inverses is fundamental. These concepts often appear in topics like transformations, where knowing both the function and its inverse aids in comprehension.
For instance, if you have a function like \(f(x) = 6x\), knowing how to find its inverse \(f^{-1}(x) = \frac{x}{6}\) illustrates how input and output roles can be switched.

• Recognizing that operations can be reversed through inverses enhances understanding of function behavior.
• In particular, precalculus involves lots of function interplay, making inverses indispensable tools in analysis.

By studying inverses, students learn about the broader implications of functions and gain skills that are vital for more advanced topics like calculus, where dealing with complex expressions and equations becomes routine.