(1). The correct option is [tex]\boxed{\bf option C}[/tex].
(2). The correct option is [tex]\boxed{\bf option C}[/tex].
Further explanation:
Concept used:
The domain of the function [tex]f(x)[/tex] is the set of all [tex]x[/tex]-values for which the function [tex]f(x)[/tex] exists as a real number and range is defined as all the corresponding values of [tex]f(x)[/tex] for the values of [tex]x[/tex] in domain.
Calculation:
Part (1)
Th function is given as follows:
[tex]f(x)=\sqrt{-x}[/tex]
In the given function a radical term is present and the argument is [tex]-x[/tex].
From the given function to be defined the argument should be always greater or equal to [tex]0[/tex].
[tex]\begin{aligned}-x\geq 0\\x\leq 0\end{aligned}[/tex]
From the above calculation it is concluded that the domain of given function is [tex](-\infty,0][/tex].
Figure 1 (attached in the end) represents the graph of the function [tex]f(x)=\sqrt{-x}[/tex].
From figure 1 it is observed that the curve of the function [tex]f(x)=\sqrt{-x}[/tex] always the above [tex]x[/tex]-axis so, it can be said that the value of the function is always greater or equal to [tex]0[/tex].
Therefore, the range of the function [tex]f(x)=\sqrt{-x}[/tex] is [tex][0,\infty)[/tex].
This implies that the correct option is [tex]\boxed{\bf option C}[/tex].
Part (2)
The function is [tex]f(x)=-\sqrt{x}[/tex].
In the given function a radical term is present and the argument is [tex]x[/tex].
From the given function to be defined the argument should be always greater or equal to [tex]0[/tex].
[tex]\boxed{x\geq 0}[/tex]
Therefore, the domain of the function [tex]f(x)=-\sqrt{x}[/tex] is [tex][0,\infty)[/tex].
Figure 2 (attached in the end) represents the graph of the function [tex]f(x)=-\sqrt{x}[/tex].
From figure 2 it is observed that the curve of the function [tex]f(x)=-\sqrt{x}[/tex] always below the [tex]x[/tex]-axis so, it can be said that the value of function always less than or equal to [tex]0[/tex].
As per the above statement it is concluded that the range of the function [tex]f(x)=-\sqrt{x}[/tex] is [tex](-\infty,0][/tex].
This implies that the correct option is [tex]\boxed{\bf option C}[/tex].
Learn more:
1. Learn more about functions https://brainly.com/question/2142762
2. Learn more about problem on numbers https://brainly.com/question/1852063
Answer details:
Grade: Senior school
Subject: Mathematics
Chapter: Function
Keywords:
Function, domain, range, corresponding value, real number, exist, domain set, range set, radical, inequality, greater than, less than, argument, square root of -x, -square root of x.
