Find the Domain f(x) = natural log of x-x^2

Problem

ƒ(x)=ln(x−x2)

Solution

1. Identify the restriction for the natural logarithm function. The argument of a logarithm must be strictly greater than zero.

x−x2>0

2. Factor the quadratic expression to find the critical points of the inequality.

x*(1−x)>0

3. Determine the roots of the expression by setting each factor to zero.

x=0

1−x=0⇒x=1

4. Test the intervals created by the roots 0 and 1 to see where the inequality x*(1−x)>0 holds true.

• For x<0 (e.g., x=−1: (−1)*(1−(−1))=−2<0 (False)

• For 0<x<1 (e.g., x=0.5: (0.5)*(1−0.5)=0.25>0 (True)

• For x>1 (e.g., x=2: (2)*(1−2)=−2<0 (False)

5. Conclude that the inequality is satisfied only when x is between 0 and 1

Final Answer

Domain: *(0,1)

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