Problem 12 - 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 \(f^{-1}(f(x))=x.\) $$f(x)=4(x-1)$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 4(x - 1)\) is \(f^{-1}(x) = (x + 4) / 4\). The properties \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\) are verified for this function and its inverse.

Step 1: Finding the Inverse Function

Let \(y = f(x)\). Therefore, the function \(f\) can be written as \(y = 4(x - 1)\). Swap the roles of \(y\) and \(x\) to obtain \(x = 4(y - 1)\). Now, solve for y to obtain the inverse function. This gives \(x = 4y - 4\). Therefore, \(4y = x + 4\), and \(y = (x + 4)/4\). Hence, the inverse function, \(f^{-1}(x)\), is \(f^{-1}(x) = (x + 4)/4\).

Step 2: Verifying the Inverse Function: \(f(f^{-1}(x)) = x\)

Substitute \(f^{-1}(x)\) into the function \(f\). Hence, \(f(f^{-1}(x)) = 4(f^{-1}(x) - 1) = 4(((x + 4)/4) - 1) = 4(x/4 + 1 - 1) = x\). This verifies that \(f(f^{-1}(x)) = x\).

Step 3: Verifying the Inverse Function: \(f^{-1}(f(x)) = x\)

Substitute \(f(x)\) into the inverse function \(f^{-1}\). Hence, \(f^{-1}(f(x)) = (f(x) + 4)/4 = ((4x - 4) + 4)/4 = x\). This verifies that \(f^{-1}(f(x)) = x\).

Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In simpler terms, it is a process where the output of one function becomes the input of another. This concept is often written as \( (f \circ g)(x) = f(g(x)) \), where \( f \) and \( g \) are functions. For our exercise, the composition is used to verify if one function is truly the inverse of another.
Here, we are asked to verify the inverse function by composing \( f \) and its proposed inverse \( f^{-1} \). If \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), then these functions are confirmed to be inverses of each other. The concept of inverse is tied heavily into composition, as the identity function \( I(x) = x \) results from a perfect inverse relationship.
Keep practicing with various functions to make this concept thoroughly understood and ingrained.

Verification of Inverse Functions

Once we have a proposed inverse function, it’s crucial to verify it to ensure our calculations are correct. This verification involves showing that when one function is applied after its inverse, the original input is retrieved. In our exercise, this means checking both \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

• This process is done using function composition, and is crucial to confirm the correctness of the inverse solution.
• For \( f(f^{-1}(x)) = x \), you start by substituting \( f^{-1} (x) \) into \( f \) and simplifying.
• Similarly, for \( f^{-1}(f(x)) = x \), substitute \( f(x) \) into \( f^{-1} \) and simplify again.

If both conditions return \( x \), the exercise is successfully verified. This rigorous check guards against mistakes in inverse calculation.

Algebraic Manipulation

Algebraic manipulation helps us solve equations and find expressions by rearranging and simplifying terms. In the task of finding an inverse function, algebraic manipulation is essential. Here’s how you can approach it:
Firstly, rewrite the function \( f(x) = 4(x - 1) \) as \( y = 4(x - 1) \). Swap \( x \) and \( y \), resulting in \( x = 4(y - 1) \). The goal is to isolate \( y \), which represents our inverse function.

• Add \( 4 \) to both sides: \( x + 4 = 4y \).
• Divide every part by \( 4 \): \( y = \frac{x + 4}{4} \).

Thus, you derive that the inverse function \( f^{-1}(x) = \frac{x + 4}{4} \).
Each step requires careful attention and systematic approach to get the expressions right. By understanding and practicing algebraic manipulation, you’ll improve your skills and accuracy in finding inverse functions.