Question

How do you find the domain and the range of $\arcsin \left( {{e}^{x}} \right)$?

Answer 1

In this problem, we have to find the domain and the range of the given inverse function. We should know that each range of an inverse function is a proper subset of the domain of the original function. We can first write the domain values for the sin function and exponential function to find the domain and the range of the given inverse function.

Complete step by step solution:
We know that the given inverse function is,
$\arcsin \left( {{e}^{x}} \right)$
We can now write the domain of the sin function.
Where, sin value $\in \left[ -1,1 \right]$.
We can now write the domain of the inverse sine function.
Where, $\arcsin \in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$
Now we have the exponential function, where ${{e}^{x}}>0$
So, we can write as,
${{e}^{x}}\in \left[ 0,1 \right]\Rightarrow x\in \left( -\infty ,0 \right]$ where $\left[ 0,1 \right]$ is the domain of the given inverse function.
we can now say that,$\arcsin \left( {{e}^{x}} \right)\in \left( 0,\dfrac{\pi }{2} \right]$, which is the range.
Therefore, Domain of the given inverse function $\arcsin \left( {{e}^{x}} \right)$ is $x\in \left( -\infty ,0 \right]$ and the range is $\left( 0,\dfrac{\pi }{2} \right]$

Note: Students make mistakes while writing the domain of the individual functions and the inverse function respectively. We should always remember that each range of an inverse function is a proper subset of the domain of the original function. We should also concentrate the brackets part as we use square brackets for which it includes the end points and we use parentheses for which we do not include the endpoints.