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

Question

James Adkins · Accepted Answer

Domain: #x in ( - oo, 0 ]#

Range: #( 0, pi/2 ]# sin value #in [ -1, 1 ], arcsin value #in [ - pi/2, pi/2 ] and #e^x > 0, and so, # e^x in [ 0, 1 ] rArr x in ( - oo, 0 ]#and #arcsin ( e^x ) in (0, pi/2 ]# See graph, depicting these aspects. graph{( y -arcsin(e^x))(y -pi/2)=0[-pi 0 0 pi/2]}

Aaliyah Abdullah · Answer

undefined 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: 
$$ \{y \in \mathbb{R} \ | \ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\} $$