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

Problem

g(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 where the expression equals zero.

x*(1−x)>0

3. Determine the roots of the inequality. The expression equals zero when x=0 and x=1 These points divide the number line into three intervals: (−∞,0) (0,1) and (1,∞)

4. Test a value from each interval to see where the inequality x*(1−x)>0 holds true.

• For x=−1 (−1)*(1−(−1))=−2<0 (False)

• For x=0.5 (0.5)*(1−0.5)=0.25>0 (True)

• For x=2 (2)*(1−2)=−2<0 (False)

5. Conclude that the inequality is satisfied only in the open interval between the roots.

Final Answer

Domain:(0,1)

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