Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x.\) $$f(x)=x+11$$

Step-by-Step Solution

Answer

The inverse function of \(f\) is \(f^{-1}(x) = x - 11\). Both equations are verified as they simplify to \(x = x\).

Step 1: Find the Inverse Function

The function is \(f(x) = x + 11\). To find the inverse, you first replace \(f(x)\) with \(y\), so the equation becomes \(y = x + 11\). Then, swap \(x\) and \(y\) to get \(x = y + 11\). Lastly, solve this equation for \(y\) to find the inverse function, \(f^{-1}(x)\). When you subtract 11 from each side, you get \(f^{-1}(x) = x - 11\).

Step 2: Test the first equation

You test \(f(f^{-1}(x)) = x\) by replacing \(f^{-1}(x)\) with the function you found in step 1, \(x - 11\). So you get \(f(x - 11) = x\). Now, you replace \(f(x)\) with the function given in the exercise \(x + 11\) to get \((x - 11) + 11 = x\). After you simplify, you are left with \(x = x\), which verifies the first equation.

Step 3: Test the second equation

You test \(f^{-1}(f(x)) = x\) by replacing \(f(x)\) with the function given in the exercise, \(x + 11\). So you get \(f^{-1}(x + 11) = x\). Now, you replace \(f^{-1}(x)\) with the function you found in step 1, \(x - 11\), to get \((x + 11) - 11 = x\). After you simplify, you are left with \(x = x\), which verifies the second equation.

Key Concepts

Function Verification

Detecting the correctness of an inverse function is a paramount part of ensuring a proper understanding of the concept. To verify an inverse function, we need to check if performing the original function and then the inverse returns the input back to its original state. This is done by using these two conditions:

• Condition 1: \(f(f^{-1}(x)) = x\)
• Condition 2: \(f^{-1}(f(x)) = x\)

Both conditions state that when a function is performed on its inverse, or vice versa, it should lead us back to the initial input value. This means, if you apply a function to its inverse, you should end up where you started.
Using these conditions helps us confirm that our inverse function truly undoes the effect of the original function. Ensuring this verification helps avoid mistakes when utilizing inverse functions in algebraic scenarios.

Inverse Operations

Inverse operations are essential in solving equations and understanding functions deeply. The concept revolves around using operations that counteract each other to simplify or solve equations. For example, addition and subtraction are inverses of each other, as are multiplication and division.
An inverse function, in this context, acts similarly in undoing what the original function has done. For the function \(f(x) = x + 11\):

• The operation involved here is addition (+11).
• To find the inverse, you use the opposite operation, which is subtraction (−11).

Inverse operations help us unravel equations and expressions, making them simpler to work with. When we apply inverse operations to a function, we aim to retrieve our original input back and verify that transformation correctness.

Algebraic Equations

Working with algebraic equations forms the basis for finding an inverse function. It involves several steps starting with representing functions in a formal mathematical manner. Let's consider the basic steps performed using the function \(f(x) = x + 11\):
Initially, the function is rewritten as an equation with \(y\) replacing \(f(x)\):

• From \(f(x) = x + 11\), it becomes \(y = x + 11\).
• Next, swap the positions of \(x\) and \(y\) to approach finding the inverse: \(x = y + 11\).
• Solving for \(y\) gives you the inverse function: \(y = x - 11\).

These algebraic maneuvers allow for the discovery of how inputs and outputs relate in mathematical functions. Through basic arithmetic operations like addition and subtraction, we manipulate and solve equations to uncover the underlying patterns in functional relationships.