Find the Domain arcsin(e^x)

Problem

domain of *arcsin(ex)

Solution

1. Identify the domain of the outer function, ƒ(u)=arcsin(u) The inverse sine function is defined only when its argument is within the interval [−1,1]

2. Set up the inequality for the inner function. For the expression to be defined, the inner function ex must satisfy the condition:

−1≤ex≤1

3. Analyze the lower bound. Since the exponential function ex is always positive for all real numbers x the condition ex≥−1 is always true.

4. Solve the upper bound inequality ex≤1 To isolate x take the natural logarithm of both sides:

ln(ex)≤ln(1)

5. Simplify the expression using logarithmic properties. Since ln(ex)=x and ln(1)=0 the inequality becomes:

x≤0

Final Answer

Domain: *(−∞,0]

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