Problem 49 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=1+\frac{1}{x} \quad \text { for } x>0 $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{1}{x-1}\), and the compositions prove true identities.

Step 1: Determine the Inverse Function

To find the inverse function, start by replacing \(f(x)\) with \(y\) such that \(y = 1 + \frac{1}{x}\). Solve for \(x\) in terms of \(y\): \(x = \frac{1}{y-1}\). Thus, the inverse function is \(f^{-1}(x) = \frac{1}{x-1}\) for \(x > 1\).

Step 2: Verify \(\left(f \circ f^{-1}\right)(x) = x\)

Substitute \(f^{-1}(x)\) into \(f(x)\): \(f(f^{-1}(x)) = f\left(\frac{1}{x-1}\right)\). Calculate: \(f\left(\frac{1}{x-1}\right) = 1 + \frac{1}{\frac{1}{x-1}} = 1 + (x - 1) = x\). This shows \((f \circ f^{-1})(x) = x\).

Step 3: Verify \(\left(f^{-1} \circ f\right)(x) = x\)

Substitute \(f(x)\) into \(f^{-1}(x)\): \(f^{-1}(f(x)) = f^{-1}\left(1 + \frac{1}{x}\right)\). Calculate: \(f^{-1}\left(1 + \frac{1}{x}\right) = \frac{1}{\left(1 + \frac{1}{x}\right) - 1} = \frac{1}{\frac{1}{x}} = x\). This confirms \((f^{-1} \circ f)(x) = x\).

Key Concepts

composition of functions

In mathematics, the idea of composing functions is like arranging words in a unique sentence. It involves combining two functions to create a new one. For example, if we have two functions, say \( f(x) \) and \( g(x) \), their composition means applying one function to the result of another. This is expressed as \( (f \circ g)(x) = f(g(x)) \). Why is this important? Imagine \( f(x) \) is about adjusting the brightness of a photo, and \( g(x) \) is about changing its size. By composing these, you can apply both effects seamlessly. In our exercise, we looked at \( f(f^{-1}(x)) \). Here, the output of the inverse function \( f^{-1}(x) \) is used as the input for \( f(x) \). It's a dance between two functions. When composed correctly, it should always bring you back to your starting point, in this case, \( x \). Some noteworthy points:- Composition order matters: \( (f \circ g)(x) \) can yield different results than \( (g \circ f)(x) \).- Composition is key in verifying inverse functions, ensuring \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).Once you are comfortable with the idea of function composition, you realize it's a foundational component for building complex mathematical models.

function verification

Verifying functions, particularly inverse functions, is like double-checking your math homework. It's essential to ensure the operations you've performed are correct. The objective is to show that when you apply a function and its inverse in sequence, you end up where you initially started. Let's dive deeper using our exercise. After finding the inverse function \( f^{-1}(x) \), we need to check that both compositions, \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \), return \( x \). This verification step confirms:

• The integrity of your inverse calculations.
• That \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\).

Verification involves a series of evaluations:- First, substitute the inverse function into the original function.- Simplify the result to see if it equals \( x \).- Then, switch their places and repeat.This two-way check ensures we didn't make any errors while calculating the inverse. A correct verification signifies that the functions perfectly "undo" each other.

algebraic manipulation

Algebraic manipulation is about bending and twisting equations until they reveal the answer. It's a set of skills and strategies to rearrange and simplify mathematical expressions. For solving our given exercise, we rely heavily on algebraic manipulations to isolate variables and simplify expressions. Think of it as organizing a cluttered desk. You're moving items around so everything falls into place. In our problem, we needed to express \( y = 1 + \frac{1}{x} \) as \( x \) in terms of \( y \). We achieved this by:

• Subtracting 1 from both sides to get \( x = \frac{1}{y-1} \).
• Understanding each step's role to ensure clarity.

These processes aren't just about reaching the correct result. They are about understanding the relationships within equations and building intuition around functions. Here are some tips for mastering this art:- Always perform equal actions on both sides of an equation.- Look to simplify complex fractions and terms as a priority.- Double-check each algebraic step to prevent missing details.The beauty of algebraic manipulation lies in its ability to directly connect equations to real-world problems, allowing us to convert unknowns into knowns with strategic planning.