Problem 36 - Calculus Volume 1

Question

For each pair of functions, find a. \(f+g\) b. \(f-g\) c. \(f \cdot g\) d. \(f / g\). Determine the domain of each of these new functions. \(\quad f(x)=3 x+4, g(x)=x-2\)

Step-by-Step Solution

Answer

a. \(4x+2\), \((- fty, fty)\); b. \(2x+6\), \((- fty, fty)\); c. \(3x^2-2x-8\), \((- fty, fty)\); d. \(\frac{3x+4}{x-2}\), \((-\infty, 2) \cup (2, \infty)\).

Step 1: Find \(f+g\)

To find \(f+g\), add the functions \(f(x)\) and \(g(x)\): \[ (f+g)(x) = f(x) + g(x) = (3x+4) + (x-2) = 3x + x + 4 - 2 = 4x + 2 \]The domain of \(f+g\) is the intersection of the domains of \(f(x)\) and \(g(x)\). Since both \(f(x)\) and \(g(x)\) are linear and defined for all real numbers, the domain of \(f+g\) is all real numbers, \((-fty, fty)\).

Step 2: Find \(f-g\)

To find \(f-g\), subtract \(g(x)\) from \(f(x)\):\[ (f-g)(x) = f(x) - g(x) = (3x+4) - (x-2) = 3x + 4 - x + 2 = 2x + 6 \]Like \(f+g\), the domain of \(f-g\) is the intersection of the domains of \(f(x)\) and \(g(x)\), which is all real numbers, \((-fty, fty)\).

Step 3: Find \(f \cdot g\)

To find \(f \cdot g\), multiply \(f(x)\) and \(g(x)\):\[ (f \cdot g)(x) = f(x) \cdot g(x) = (3x+4)(x-2) \]Distribute and simplify:\[ (f \cdot g)(x) = 3x \cdot x - 3x \cdot 2 + 4 \cdot x - 4 \cdot 2 = 3x^2 - 6x + 4x - 8 = 3x^2 - 2x - 8 \]The domain of \(f \cdot g\) is all real numbers, \((-fty, fty)\), as it is based on multiplication of polynomials.

Step 4: Find \(f / g\)

To find \(f / g\), divide \(f(x)\) by \(g(x)\):\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{3x + 4}{x - 2} \]Here, the domain will be all real numbers except where the denominator equals zero. Thus, set \(x-2 = 0\), which gives \(x = 2\). So, the domain of \(f / g\) is all real numbers except \(x = 2\), \((-\infty, 2) \cup (2, \infty)\).

Key Concepts

Function Addition

When adding two functions, you simply add the output of the two functions for the same input. This process involves combining the terms of each function. For example, let's consider the functions:

• \( f(x) = 3x + 4 \)
• \( g(x) = x - 2 \)

To find \( (f+g)(x) \), you add these two functions: \[ (f+g)(x) = (3x + 4) + (x - 2) = 4x + 2 \] Each term in \( f(x) \) and \( g(x) \) is combined together, simplifying to give the resulting expression. Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain for \( f+g \) is also all real numbers, which means it extends from negative infinity to positive infinity, \((-fty, fty)\). Understanding the concept of function addition is crucial, as it frequently appears in algebraic manipulations and functions' analysis.

Function Subtraction

Subtraction of two functions involves taking away the output of one function from the other for the same input value. Let's illustrate with our example functions:

• \( f(x) = 3x + 4 \)
• \( g(x) = x - 2 \)

To find \( (f-g)(x) \), subtract \( g(x) \) from \( f(x) \): \[ (f-g)(x) = (3x + 4) - (x - 2) = 2x + 6 \] In this process, you distribute the negative sign across all terms of \( g(x) \), then combine like terms to simplify. Because both functions are linear and cover all real numbers, their domain remains unchanged as real numbers, symbolically \((-fty, fty)\). Function subtraction can be a helpful tool in contexts like solving equations or determining differences in functional outputs.

Function Multiplication

Multiplying two functions requires multiplying their outputs for the same input value. For our example functions:

• \( f(x) = 3x + 4 \)
• \( g(x) = x - 2 \)

The multiplication is set up as follows:\[ (f \cdot g)(x) = (3x + 4)(x - 2) \] Distribute each term in the first expression across each term in the second expression, combining like terms:\[ (f \cdot g)(x) = 3x^2 - 2x - 8 \] In this case, like when working with polynomials, distribute and simplify by gathering similar terms together. The domain remains all real numbers, due to the operations of multiplication across all real combinations \((-fty, fty)\). Function multiplication can expand or contract function ranges and is especially useful in combined transformation applications.

Function Division

For function division, the operation is to divide the output of one function by the other at each input point, taking great care to avoid dividing by zero. Consider our functions:

• \( f(x) = 3x + 4 \)
• \( g(x) = x - 2 \)

You perform the division as:\[ \left( \frac{f}{g} \right)(x) = \frac{3x + 4}{x - 2} \] The domain for division requires excluding any value that makes the denominator zero, because division by zero is undefined. So, solving \( x - 2 = 0 \), we find \( x = 2 \). Thus, the domain is all real numbers except \( x = 2 \), or \((-fty, 2) \cup (2, fty)\). Function division is often used to solve ratio problems and to study the behavior of functions near problematic points. Such analysis is critical for proper understanding and application of rational functions.