Problem 14 - Discrete Mathematics with Applications

Question

Determine if the function \(g\) is the inverse of the corresponding function \(f\). $$f(x)=x^{2}, x \leq 0 ; g(x)=-\sqrt{x}, x \geq 0$$

Step-by-Step Solution

Answer

The inverse function of \(f(x) = x^2\), considering the domain restriction \(x \leq 0\), is found to be \(f^{-1}(x) = -\sqrt{x}\). Since this matches the given function \(g(x) = -\sqrt{x}\) and their domain restrictions are compatible, function \(g\) is indeed the inverse of the corresponding function \(f\).

Step 1: Find the inverse of function \(f\)

To find the inverse function of "\(f(x) = x^2\)", we switch the roles of \(x\) and \(y\) and solve for \(y\). This will give us the inverse function. $$y=x^2$$ $$x=y^2$$ Solve for y: $$y = \pm \sqrt x$$ Since \(f(x)\) has the condition that \(x \leq 0\), we consider the negative square root only. Therefore the inverse function for \(f(x)\) is \(f^{-1}(x)=-\sqrt x\).

Step 2: Compare the function \(g\) to the inverse \(f^{-1}\)

Now we compare the function \(g(x)=-\sqrt x\) to the inverse of function \(f\), \(f^{-1}(x)=-\sqrt x\). Since \(g(x) = f^{-1}(x) = -\sqrt{x}\) and the domain restrictions for both functions are compatible (\(x \leq 0\) for \(f(x)\) and \(x \geq 0\) for \(g(x)\)), the function \(g\) is indeed the inverse of the corresponding function \(f\).

Key Concepts

Discrete Mathematics and Inverse Functions

Discrete mathematics plays a pivotal role in understanding the concept of inverse functions. When dealing with functions, especially in the realm of discrete mathematics, a function is seen as a special type of relation between two sets that associates each element of the first set with exactly one element of the second set.

An inverse function essentially 'reverses' the assignment made by the original function. To find the inverse of a function in discrete mathematics, one would typically swap the input and output variables and solve for the new output. What makes this a part of discrete mathematics is the condition that functions and their inverses often deal with discrete values or operate within a discrete domain.

In the case of determining if one function is the inverse of another, we must ensure that the original function is bijective, meaning it is both injective (one-to-one) and surjective (onto). This is vital because only bijective functions have inverses that are functions themselves. In this exercise, the given function is defined in such a manner that it satisfies these conditions within the given domain, allowing us to properly compute its inverse.

Function Operations

Function operations are procedures we can perform on functions, including addition, subtraction, multiplication, division, and composition. In the context of finding an inverse function, a specific type of operation is conducted: the inverse operation. The inverse operation seeks to reverse the original function's effect.

When we talk about function operations concerning inverses, we are concerned with the composition of a function and its inverse. The composition must satisfy two conditions:

• If we compose a function with its inverse, we should get the identity function on the domain of the original function: \( f(f^{-1}(x)) = x \) for all x in the domain of \(f^{-1}\).
• If we compose the inverse of a function with the function itself, we should get the identity function on the domain of the inverse: \( f^{-1}(f(x)) = x \) for all x in the domain of \(f\).

The step-by-step solution for the exercise demonstrates this relationship clearly by comparing function \(g\) to the inverse function derived from function \(f\), highlighting that \(g(x)\) must perform the exact 'undo' operation that \(f\)'s inverse would.

Square Root Functions

Square root functions are a specific type of mathematical function involving the square root of a variable. These functions are defined as \( f(x) = \sqrt{x} \), but in the context of inverse functions, we often deal with the negative square root to find the inverse of a squared function with a restricted domain. This is especially relevant when the original function is not strictly increasing or decreasing over its entire range.

In dealing with the inverse of a square function \( f(x) = x^2 \), where \( x \) is restricted to negative values, the square root function that serves as its inverse will naturally be the negative square root. This is because only the negative square root will produce the appropriate bijective mapping back to the original function's negative domain.

Thus, the solution shows that in order to maintain the proper domain and range that correlates with \(f(x)\) and ensures \(f\) and \(g\) are indeed inverses of one another, \(g(x)\) must be defined as \(g(x) = -\sqrt{x} \) for \( x \geq 0 \). This restricts the function to yielding only non-positive outputs, aligning with the input restriction of \(f\).