In Exercises 31–50, find f/g and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the function \( \frac{f}{g} \) where \( f(x) = x - 5 \) and \( g(x) = 3x^2 \).
2. Step 2: Write the expression for \( \frac{f}{g} \). This is \( \frac{f(x)}{g(x)} = \frac{x - 5}{3x^2} \).
3. Step 3: Simplify the expression if possible. In this case, there are no common factors to simplify between the numerator and the denominator.
4. Step 4: Determine the domain of \( \frac{f}{g} \). The domain of a rational function is all real numbers except where the denominator is zero.
5. Step 5: Set the denominator equal to zero and solve for \( x \). Solve \( 3x^2 = 0 \) to find the values of \( x \) that are not in the domain. The solution is \( x = 0 \). Therefore, the domain is all real numbers except \( x = 0 \).

Key Concepts

Function Division

Function division involves creating a new function by dividing one function by another. In this case, f/g means taking the function f(x) = x - 5 and dividing it by g(x) = 3x². The resulting function will be expressed as (x - 5) / (3x²), which is essential for further analysis.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f/g, we must identify values of x that do not make the denominator zero, as division by zero is undefined. Thus, we need to solve the equation 3x² = 0 to find any restrictions on the domain.

Finding Restrictions

Finding restrictions involves determining the values that must be excluded from the domain of a function. In this case, we set the denominator g(x) = 3x² equal to zero to find x = 0. Therefore, the domain of the function f/g excludes x = 0, leading to the domain being all real numbers except zero.