To determine the domain and range of the function \( w(x) = -(3x)^{\frac{1}{2}} - 4 \), let's break it down step by step.
1. Domain:
- The expression inside the square root, \( 3x \), must be non-negative because you cannot take the square root of a negative number in the reals.
- Therefore, we set up the inequality \( 3x \geq 0 \).
- Solving for \( x \), we get \( x \geq 0 \).
Hence, the domain is \( x \geq 0 \).
2. Range:
- Next, consider the range of the function.
- The square root function, \( (3x)^{\frac{1}{2}} \), produces non-negative values (ranging from 0 to \(\infty\)).
- Multiplying by -1, \( -(3x)^{\frac{1}{2}} \), will result in non-positive values (ranging from 0 to \(-\infty\)).
- Finally, by subtracting 4, \( -(3x)^{\frac{1}{2}} - 4 \) will shift this down by 4 units, resulting in values ranging from -4 to \(-\infty\).
Hence, the range is \( w(x) \leq -4 \).
So, filling in the blanks:
- Domain: \( x \geq 0 \)
- Range: [tex]\( w(x) \leq -4 \)[/tex]
