Find the Domain

Problem

Domain of (x2−81)/(3*x−18)/(x2+18*x+81)/(x2+3*x−54)

Solution

1. Identify all denominators in the complex fraction that cannot be zero.

2. Set the denominator of the top fraction to not equal zero: 3*x−18≠0

3. Solve for x 3*x≠18⇒x≠6

4. Set the denominator of the bottom fraction to not equal zero: x2+3*x−54≠0

5. Factor the quadratic: (x+9)*(x−6)≠0⇒x≠−9 and x≠6

6. Identify that the entire divisor (the bottom fraction) cannot be zero because division by zero is undefined.

7. Set the numerator of the bottom fraction to not equal zero: x2+18*x+81≠0

8. Factor the perfect square trinomial: (x+9)2≠0⇒x≠−9

9. Combine all restrictions found: x≠6 and x≠−9

Final Answer

Domain: *{[x∈ℝ,x≠6,x≠−9]}

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