Find the Domain

Problem

y=ln(x2)+√(,x)+arctan((ex)/x)+1/x

Solution

1. Analyze the natural logarithm ln(x2) The argument of a logarithm must be strictly positive.

x2>0

⇒x≠0

2. Analyze the square root √(,x) The radicand of an even root must be non-negative.

x≥0

3. Analyze the fraction 1/x and the argument of the arctangent (ex)/x The denominator of any fraction cannot be zero.

x≠0

4. Analyze the arctangent function arctan(u) The domain of the arctangent function is all real numbers, so it imposes no additional constraints on its argument beyond the division by zero already identified.

u∈(−∞,∞)

5. Combine all constraints to find the intersection of the individual domains. We require x≠0 and x≥0

x>0

Final Answer

Domain:(0,∞)

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