Find the Domain 1/(3x^2-27)

Problem

1/(3*x2−27)

Solution

1. Identify the condition for the domain of a rational function, which requires that the denominator cannot be equal to zero.

3*x2−27≠0

2. Factor out the greatest common factor from the quadratic expression to simplify the equation.

3*(x2−9)≠0

3. Apply the difference of squares formula, a2−b2=(a−b)*(a+b) to the expression inside the parentheses.

3*(x−3)*(x+3)≠0

4. Solve for the values of x that make the denominator zero by setting each factor to zero.

x−3=0⇒x=3

x+3=0⇒x=−3

5. State the domain by excluding these specific values from the set of all real numbers.

Domain={x|x≠3,x≠−3}

Final Answer

Domain=(−∞,−3)∪(−3,3)∪(3,∞)

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