To find the domain and range of the function [tex]\( w(x) = -(3x)^{\frac{1}{2}} - 4 \)[/tex], let's analyze it step-by-step.
### Domain:
1. The function involves the square root [tex]\( (3x)^{\frac{1}{2}} \)[/tex]. For the square root to be defined and real, the expression inside the square root must be non-negative:
[tex]\[
3x \geq 0
\][/tex]
2. Solving this inequality for [tex]\( x \)[/tex]:
[tex]\[
x \geq 0
\][/tex]
So, the domain of the function [tex]\( w(x) \)[/tex] is:
[tex]\[
x \geq 0
\][/tex]
### Range:
1. The minimum value of the square root function [tex]\( (3x)^{\frac{1}{2}} \)[/tex] is 0 since the square root of 0 is 0.
2. Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
w(0) = -(3 \cdot 0)^{\frac{1}{2}} - 4 = -4
\][/tex]
3. As the value of [tex]\( x \)[/tex] increases, the term [tex]\( (3x)^{\frac{1}{2}} \)[/tex] increases as well.
4. Since this term is multiplied by -1 in the function, the overall value of [tex]\( -(3x)^{\frac{1}{2}} \)[/tex] decreases, making [tex]\( w(x) \)[/tex] take on values less than or equal to -4.
So, the range of the function [tex]\( w(x) \)[/tex] is:
[tex]\[
w(x) \leq -4
\][/tex]
Thus, the correct answers are:
[tex]\[
\text{domain: } x \geq 0
\][/tex]
[tex]\[
\text{range: } w(x) \leq -4
\][/tex]
