# How do you find the domain and range of #arcsin(e^x)#?

Domain:

Range:

and so,

See graph, depicting these aspects.

graph{( y -arcsin(e^x))(y -pi/2)=0[-pi 0 0 pi/2]}

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To find the domain and range of arcsin(e^x):

Domain: The domain of arcsin(e^x) is the set of all real numbers x such that e^x lies within the range of arcsine function, which is [-1, 1]. Therefore, the domain is all real numbers x such that e^x is between -1 and 1, inclusive. In mathematical notation, the domain is given by: [ {x \in \mathbb{R} \ | \ -1 \leq e^x \leq 1} ]

Range: The range of arcsin(e^x) is the set of all real numbers y such that arcsin(y) is defined. Since the range of the exponential function e^x is (0, ∞), the range of arcsin(e^x) would be limited by the range of the arcsine function, which is [-π/2, π/2]. Therefore, the range of arcsin(e^x) is [-π/2, π/2]. In mathematical notation, the range is given by: [ {y \in \mathbb{R} \ | \ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}} ]

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