Problem 1 - A Book of Abstract Algebra

Question

Each of the following functions \(f\) is bijective. Describe its inverse. \(f:(0, \infty) \rightarrow(0, \infty)\), defined by \(f(x)=1 / x\)

Step-by-Step Solution

Answer

The inverse function is defined by \( f^{-1}(y) = \frac{1}{y} \) for \( y > 0 \).

Step 1: Understanding the Function

The given function is \( f(x) = \frac{1}{x} \), which is defined for all \( x > 0 \). This function maps positive real numbers to positive real numbers. The goal is to determine the function that 'undoes' this action, or the inverse function.

Step 2: Define the Inverse Function

To find the inverse function, we need to express \( x \) in terms of \( y \) where \( y = f(x) \). This means solving the equation \( y = \frac{1}{x} \) for \( x \).

Step 3: Solve for \( x \)

Starting with the equation \( y = \frac{1}{x} \), multiply both sides by \( x \) to get \( yx = 1 \). Then, solve for \( x \) by dividing both sides by \( y \), yielding \( x = \frac{1}{y} \).

Step 4: Express the Inverse Function

The inverse function of \( f \), denoted as \( f^{-1} \), is thus \( f^{-1}(y) = \frac{1}{y} \) for all \( y > 0 \).

Step 5: Confirmation of Inverses

To confirm, check that \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(y)) = f\left(\frac{1}{y}\right) = y \). For \( f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) = x \). These confirm that \( f^{-1} \) is indeed the inverse.

Key Concepts

Bijective Functions

In mathematics, bijective functions are one of the foundational concepts when it comes to understanding functions and their inverses. A function is considered bijective if it is both injective (one-to-one) and surjective (onto). This means:

• Each element in the function's domain maps to a unique element in the codomain.
• Every element in the codomain is mapped by some element from the domain.

A simple way to think about this is that a bijective function forms a perfect pairing between the domain and the codomain where there are no leftover elements in either.
For the function in our exercise, given as \( f(x) = \frac{1}{x} \) with a domain and codomain of positive real numbers \((0, \infty)\), it is bijective. Each positive number maps to exactly one other positive number, ensuring every possible y-value is covered, fulfilling the conditions of bijection.

Solving Equations

Solving equations is an essential skill when dealing with functions and especially when finding their inverses. In our exercise, to find the inverse function, we needed to solve the equation \( y = \frac{1}{x} \) for \( x \). This required us to:

• First multiply both sides by \( x \) to clear the fraction, giving us \( yx = 1 \).
• Then, divide both sides by \( y \) to solve for \( x \), resulting in \( x = \frac{1}{y} \).

Through these steps, we effectively expressed \( x \) in terms of \( y \), which allowed us to build the inverse function. Solving equations like this is fundamental as it transforms dependent relationships, reversing the original function to uncover its inverse.

Function Inverses

The inverse of a function essentially "reverses" the operation of the original function. If a function \( f \) takes input \( x \) and turns it into \( y \), its inverse \( f^{-1} \) takes \( y \) and turns it back into \( x \). This symmetry is crucial in understanding the relationship between functions and their inverses.
For a function to have an inverse, it must be bijective. Once confirmed, the inverse is found by expressing \( x \) in terms of \( y \) from the equation \( y = f(x) \). In our exercise, the function \( f(x) = \frac{1}{x} \) had the inverse \( f^{-1}(y) = \frac{1}{y} \). Validating this:

• \( f(f^{-1}(y)) = f\left(\frac{1}{y}\right) = y \)
• \( f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) = x \)

Both expressions confirming the inverse function's accuracy.

Algebraic Proofs

Algebraic proofs involve using algebra to rigorously demonstrate the truth of a mathematical statement. In the context of our exercise, two important algebraic proofs verify that we have correctly identified the inverse function.
Firstly, we check the composition \( f(f^{-1}(y)) \):

• By substituting the inverse into the original function, \( f(f^{-1}(y)) = f\left(\frac{1}{y}\right) \), we simplify to \( y \), confirming that applying \( f \) to its inverse yields the original y-value.

Secondly, we verify \( f^{-1}(f(x)) \):

• Substitute the function into its inverse, \( f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) \), simplifies to \( x \), showing that applying the inverse to the function returns the original x-value.

These proofs ensure we have indeed found the correct inverse function, providing confidence in our algebraic manipulations and logical deductions.