Find the Domain f(x) = fourth root of x^2+3x

Problem

ƒ(x)=√(4,x2+3*x)

Solution

1. Identify the restriction for an even root. For the function to be defined in the real number system, the radicand (the expression inside the root) must be greater than or equal to zero.

x2+3*x≥0

2. Factor the quadratic expression to find the critical points where the expression equals zero.

x*(x+3)≥0

3. Determine the critical points by setting each factor to zero.

x=0

x+3=0⇒x=−3

4. Test the intervals created by the critical points x=−3 and x=0 to see where the inequality holds true.
   For x<−3 (e.g., x=−4: (−4)*(−4+3)=(−4)*(−1)=4≥0 (True)
   For −3<x<0 (e.g., x=−1: (−1)*(−1+3)=(−1)*(2)=−2≥0 (False)
   For x>0 (e.g., x=1: (1)*(1+3)=(1)*(4)=4≥0 (True)

5. Combine the intervals where the inequality is satisfied. The domain includes the points where the expression is positive or zero.

(−∞,−3]∪[0,∞)

Final Answer

Domain:(−∞,−3]∪[0,∞)

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