Problem 46 - Precalculus Essentials

Question

Find \(f+g, f-g,\) fg, and \(\frac{f}{x}\). Determine the domain for each function. $$f(x)=\frac{9 x}{x-4}, g(x)=\frac{7}{x+8}$$

Step-by-Step Solution

Answer

The functions after the operations are: \(f(x) + g(x) = \frac{9x^2 + 91x - 112}{x^2 + 4x - 32}\); \(f(x) - g(x) = \frac{9x^2 + 21x - 112}{x^2 + 4x - 32}\); \(f(x) * g(x) = \frac{63x}{x^2 + 4x - 32}\) and \(\frac{f(x)}{x} = \frac{9}{x-4}\). The domain of f(x) is all real numbers except 4 and the domain of g(x) is all real numbers except -8.

Step 1: Analyze the functions

We have two functions \(f(x)=\frac{9 x}{x-4}\) and \(g(x)=\frac{7}{x+8}\). It is important to ensure that the denominator of these fraction functions does not equal to zero, as that would cause the function to be undefined. This condition will help us establish the domain of each function.

Step 2: Perform the Operation - Addition

Add the two functions together to get the function \(h(x) = f(x) + g(x)\). Therefore, \(h(x) = \frac{9 x}{x-4} + \frac{7}{x+8}\). To add these fractions, find a common denominator by multiplying the two denominators together, and then adjust the numerators accordingly. This results in \(h(x)= \frac{9x^2 + 56x + 7x(x-4)}{(x-4)(x+8)}\). Simplifying yields \(h(x) = \frac{9x^2 + 63x + 28x - 112}{x^2 + 4x - 32}\). Further simplifying this to \(h(x)= \frac{9x^2 + 91x - 112}{x^2 + 4x - 32}\).

Step 3: Perform the Operation - Subtraction

Next, subtract g(x) from f(x) to get the function \(h(x) = f(x) - g(x)\). Which results in \(h(x) = \frac{9 x}{x-4} - \frac{7}{x+8}\). Similar to the addition operation, find a common denominator and simplify the expression, yielding \(h(x)= \frac{9x^2 + 56x - 7x(x-4)}{(x-4)(x+8)}\), which further simplifies to \(h(x)= \frac{9x^2 + 21x - 112}{x^2 + 4x - 32}\).

Step 4: Perform the Operation - Multiplication

Multiply the two functions together to get the function \(h(x) = f(x) * g(x) \). Therefore \(h(x) = \frac{9 x}{x-4} * \frac{7}{x+8} = \frac{63x}{x^2 + 4x - 32}\).

Step 5: Perform the operation - Division and domain determination

Divide f(x) by x to get the function \(h(x) = \frac{f(x)}{x} \). Therefore \(h(x) = \frac{\frac{9 x}{x-4}}{x}=\frac{9}{x-4}\). Determine the domain for each of these functions by setting the denominator equal to zero, and solving for x. For f(x), \(x-4 ≠ 0\) which implies \(x ≠ 4\). For g(x), \(x+8 ≠ 0\) which implies \(x ≠ -8\). Therefore, the domain of f(x) is all real numbers except 4, and the domain of g(x) is all real numbers except -8.

Key Concepts

Domain of Functions

Understanding the domain of a function is crucial because it tells you all the possible inputs (values of \(x\)) that the function can take without breaking any rules, like dividing by zero. In the given problem, the function \(f(x) = \frac{9x}{x-4}\) has a denominator of \(x-4\), which means we cannot plug in \(x=4\) because it would make the denominator zero, leading to an undefined expression.
Similarly, \(g(x) = \frac{7}{x+8}\) has a restriction where \(x = -8\) is not allowed. Therefore, the domains of these functions are:

• For \(f(x)\), all real numbers except \(x = 4\).
• For \(g(x)\), all real numbers except \(x = -8\).

When performing operations like addition, subtraction, multiplication, or division, always check if the domains of the resulting functions expand or remain the same by reassessing what would make any denominator zero.

Function Addition and Subtraction

Adding and subtracting functions follows similar rules to adding and subtracting numbers, but involves dealing with their domains and possibly with rational expressions. To add the functions \(f(x)\) and \(g(x)\), we need a common denominator and simplify the resultant fraction.
For example, when

• Adding: \(h(x) = f(x) + g(x) = \frac{9x}{x-4} + \frac{7}{x+8}\),

we first find a common denominator \((x-4)(x+8)\). The numerators are then adjusted and the expression simplifies to \(h(x) = \frac{9x^2 + 91x - 112}{(x-4)(x+8)}\).
Similarly, subtraction involves the same common denominator and results in \(h(x) = \frac{9x^2 + 21x - 112}{(x-4)(x+8)}\).
In both operations, always double-check to see if any new restrictions impact the domains, such as additional factors in the denominator.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions require special attention to simplify and manipulate. They are particularly susceptible to restrictions in their domains, typically because of potential division by zero.
In the exercise, both functions \(f(x)\) and \(g(x)\) are rational expressions. When dealing with operations such as multiplication and division, it is important to keep track of these expressions.

• Multiplying: \(f(x) \cdot g(x) = \frac{9x}{x-4} \cdot \frac{7}{x+8} = \frac{63x}{(x-4)(x+8)}\).
• Dividing by \(x\) leads to an alternate form of \(f(x)\), \(h(x) = \frac{f(x)}{x} = \frac{9x}{x(x-4)}\), requiring another look at the domain.

Carefully simplifying these rational expressions while ensuring no extra restrictions on the domain arise is key. By understanding these expressions and their characteristics, you become more adept at managing similar problems in algebra.