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. You are tasked with finding the quotient of two functions, f(x) and g(x), denoted as (f/g)(x), and determining the domain of the resulting function.
2. Step 2: Write the formula for the quotient of two functions. The quotient is defined as (f/g)(x) = f(x) / g(x). Substitute the given functions into this formula: (f/g)(x) = (x - 5) / (3x²).
3. Step 3: Analyze the domain of the function. The domain of (f/g)(x) includes all values of x for which g(x) ≠ 0, because division by zero is undefined. Set g(x) = 3x² ≠ 0 and solve for x. This implies x ≠ 0.
4. Step 4: Combine the domain restrictions. Since g(x) is a polynomial, it is defined for all real numbers except where g(x) = 0. Therefore, the domain of (f/g)(x) is all real numbers except x = 0.
5. Step 5: Write the final expression for (f/g)(x) and state the domain. The quotient function is (f/g)(x) = (x - 5) / (3x²), and 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 refers to 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, which is crucial for ensuring the function is valid and defined.