When we consider the expression \(\sqrt{-x}\), it is important to recognize that we are operating within the context of complex numbers rather than just real numbers. The square root of a negative number is not defined within the real number system because no real number, when squared, gives a negative result. However, in the set of complex numbers, we introduce the imaginary unit \(i\) where \(i^2 = -1\).
### Understanding \(\sqrt{-x}\)
For the square root of a negative number \(-x\) (assuming \(x > 0\) for the sake of discussion), we can rewrite it using the imaginary unit:
[tex]\[
\sqrt{-x} = \sqrt{x \cdot (-1)} = \sqrt{x} \cdot \sqrt{-1} = \sqrt{x} \cdot i
\][/tex]
Thus, the expression \(\sqrt{-144}\) can be calculated as follows:
[tex]\[
\sqrt{-144} = \sqrt{144} \cdot i = 12i
\][/tex]
### Determining the Domain
The domain of the function \(f(x) = \sqrt{-x}\) refers to the set of \(x\)-values for which the function is defined. To ensure that the expression under the square root is non-negative (because we cannot take the square root of a positive number under the context of the real part of the function), \( -x \) must be non-negative:
[tex]\[
-x \geq 0 \implies x \leq 0
\][/tex]
Therefore, the domain of \( f(x) = \sqrt{-x} \) is all real numbers \(x\) such that \( x \leq 0 \).
### Determining the Range
The range of a function is the set of all possible values of the function. For \(f(x) = \sqrt{-x}\):
- If we consider \( x = 0 \):
[tex]\[
f(0) = \sqrt{-0} = 0
\][/tex]
- If we consider negative values of \(x\), say \(x = -a\) where \( a > 0 \):
[tex]\[
f(-a) = \sqrt{-(-a)} = \sqrt{a}
\][/tex]
Thus, the output of \( f(x) = \sqrt{-x} \) will always be non-negative real numbers (since the square root function produces non-negative results). Therefore, the range of \( f(x) = \sqrt{-x} \) is all non-negative real numbers.
### Summary
To summarize:
- The domain of \( f(x) = \sqrt{-x} \) is \( x \leq 0 \).
- The range of \( f(x) = \sqrt{-x} \) is all non-negative real numbers.
These conclusions allow us to understand how the function [tex]\( \sqrt{-x} \)[/tex] is defined and how it behaves within the realm of real numbers, acknowledging the specific contexts where the imaginary unit is involved.
