Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \((f+g)(x),\) (b) \((f-g)(x)\) , (c) \((f g)(x),\) and \((d)(f / g)(x) .\) What is the domain of \(f / g ?\) $$f(x)=\sqrt{x^{2}-4}, \quad g(x)=\frac{x^{2}}{x^{2}+1}$$

Step-by-Step Solution

Answer

\((f+g)(x) = \sqrt{x^{2}-4} + \frac{x^{2}}{x^{2}+1}\), \((f-g)(x) = \sqrt{x^{2}-4} - \frac{x^{2}}{x^{2}+1}\), \((fg)(x) = \left(\sqrt{x^{2}-4}\right) \cdot \left(\frac{x^{2}}{x^{2}+1}\right), \((f/g)(x) = \frac{\sqrt{x^{2}-4}}{\frac{x^{2}}{x^{2}+1}\) with the domain as \([- \infty, -2] \cup [2, \infty]\).

Step 1: Function Addition

First, find \((f+g)(x)\). This is simply the sum of the two provided functions, \(f(x)\) and \(g(x)\). \((f+g)(x) = f(x) + g(x) = \sqrt{x^{2}-4} + \frac{x^{2}}{x^{2}+1}\)

Step 2: Function Subtraction

Next, find \((f-g)(x)\). This is the difference of the two provided functions, \(f(x)\) and \(g(x)\). \((f-g)(x) = f(x) - g(x) = \sqrt{x^{2}-4} - \frac{x^{2}}{x^{2}+1}\)

Step 3: Function Multiplication

Thirdly, find \((fg)(x)\). This is the multiplication of the provided functions, \(f(x)\) and \(g(x)\). \((fg)(x) = f(x) \cdot g(x) = \left(\sqrt{x^{2}-4}\right) \cdot \left(\frac{x^{2}}{x^{2}+1}\right)\)

Step 4: Function Division and Domain

Find \((f / g)(x)\). This is the result of dividing \(f(x)\) by \(g(x)\). \((f / g)(x) = f(x) / g(x) = \frac{\sqrt{x^{2}-4}}{\frac{x^{2}}{x^{2}+1}}\). Finally, we need to find the domain of \(f / g(x)\). The domain is all real numbers except those for which \(g(x) = 0\). However, g(x) is never equal to zero, as \(x^{2}+1 > 0)\) for all real numbers \(x\). Therefore, the domain of \(f/g(x)\) is the domain of \(f(x)\), which is all \(x\) such that \(x^{2}-4 \geq 0\). Thus, the domain of \(f/g(x)\) is \([- \infty, -2] \cup [2, \infty]\)

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial for working with any function-related operations. The domain essentially refers to all the possible 'input' values that a function can accept. For example, with the function \(f(x)=\sqrt{x^{2}-4}\), the domain is restricted to values of \(x\) for which the expression under the square root is non-negative, since we can't take the square root of a negative number in real numbers.

Looking at \(x^{2}-4\), we set it greater than or equal to zero to find the domain. Solving \(x^{2}-4 \geq 0\) yields \(x \leq -2\) or \(x \geq 2\), meaning the domain of \(f(x)\) is \(x \in [- \infty, -2] \cup [2, \infty]\). It's important to remember that when a function is divided by another function, the domain is all the values where the denominator is not equal to zero as well as where the numerator is defined.

Function Addition

When we add two functions, like \(f(x)\) and \(g(x)\), we perform what is known as function addition. This operation is quite straightforward—it simply involves adding the values of \(f(x)\) and \(g(x)\) for any \(x\) within their domains.

For our specific functions, \(f(x) = \sqrt{x^{2}-4}\) and \(g(x) = \frac{x^{2}}{x^{2}+1}\), the addition looks like this: \[ (f+g)(x) = f(x) + g(x) = \sqrt{x^{2}-4} + \frac{x^{2}}{x^{2}+1} \]. When writing the answer, ensure to simplify as much as possible. However, the domain for this new function will be the intersection of the domains of \(f\) and \(g\).

Function Subtraction

Function subtraction follows a similar logic to function addition, except we subtract the values of \(g(x)\) from \(f(x)\). It's like removing \(g\)'s effect from \(f\).

For \(f(x) - g(x)\), using our example functions, we have \[ (f-g)(x) = f(x) - g(x) = \sqrt{x^{2}-4} - \frac{x^{2}}{x^{2}+1} \]. This operation may change the shape and properties of the original function, but it doesn't affect the domain as long as \(g(x)\) is not undefined for any \(x\) within the domain of \(f(x)\).

Function Multiplication

Multiplying functions combines the 'effects' of both functions into a single new function. In mathematical terms, \( (fg)(x) = f(x) \cdot g(x)\). Here, the multiplication operation is performed on the output values of both functions.

For our textbook example functions, we will have \[ (fg)(x) = \left(\sqrt{x^{2}-4}\right) \cdot \left(\frac{x^{2}}{x^{2}+1}\right) \]. Simplification is again key to finding the most compact form of the answer. The domain of the resulting function is the intersection of the domains of both \(f\) and \(g\) since we need valid inputs for both functions to perform the multiplication.

Function Division

Function division, much like traditional division, involves dividing the output of one function by the output of another. Formally, \( (f / g)(x) = \frac{f(x)}{g(x)} \), assuming \(g(x) \eq 0\). The division of two functions can significantly alter the behavior and domain of the resulting function.

For the given exercise, we divide \(f(x)\) by \(g(x)\) as follows: \[ (f / g)(x) = \frac{\sqrt{x^{2}-4}}{\frac{x^{2}}{x^{2}+1}} \]. This result can often be simplified further. In this case, the domain is dictated by the domain of \(f(x)\) as well as the requirement that \(g(x)\) does not equal zero. Since \(g(x)\) is never zero for all real numbers \(x\), the limiting factor for the domain is solely based on the domain of \(f(x)\), which is \(x \in [- \infty, -2] \cup [2, \infty]\).