Find the Domain square root of (x^2+7x+10)/(x+3)

Problem

√(,(x2+7*x+10)/(x+3))

Solution

1. Identify the constraints for the domain. For a square root function √(,ƒ(x)) the radicand must be non-negative: (x2+7*x+10)/(x+3)≥0 Additionally, the denominator cannot be zero: x+3≠0

2. Factor the numerator of the expression. The quadratic x2+7*x+10 factors into (x+5)*(x+2)

3. Determine the critical points where the expression is zero or undefined. These occur at x=−5 x=−2 and x=−3

4. Test the intervals created by the critical points: (−∞,−5] [−5,−3) (−3,−2] and [−2,∞)

5. Evaluate the sign of ((x+5)*(x+2))/(x+3) in each interval.

• For x∈(−∞,−5] the expression is ≤0

• For x∈[−5,−3) the expression is ≥0

• For x∈(−3,−2] the expression is ≤0

• For x∈[−2,∞) the expression is ≥0

6. Exclude x=−3 because it makes the denominator zero, resulting in an undefined expression.

7. Combine the intervals where the expression is non-negative.

Final Answer

Domain:[−5,−3)∪[−2,∞)

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