Question

type the correct answer in each box. use numerals instead of words. the function $f(x) = x^{\frac{1}{2}}$ is transformed to get function $w$. $w(x) = -(3x)^{\frac{1}{2}} - 4$ what are the domain and the range of function $w$? domain: $x \geq \square$ range: $w(x) \leq \square$

Explanation:

Step1: Find the domain of \( w(x) \)

The function \( w(x) = -(3x)^{\frac{1}{2}} - 4 \) involves a square root (since \( (3x)^{\frac{1}{2}}=\sqrt{3x} \)). For the square root to be defined, the expression inside the square root must be non - negative. So we set \( 3x\geq0 \).
Dividing both sides of the inequality \( 3x\geq0 \) by 3 (since 3 is a positive number, the direction of the inequality remains the same), we get \( x\geq0 \).

Step2: Find the range of \( w(x) \)

First, consider the function \( y=(3x)^{\frac{1}{2}}=\sqrt{3x} \). The range of \( \sqrt{3x} \) is \( y\geq0 \) because the square root of a non - negative number is non - negative.
Now, for the function \( w(x)=-(3x)^{\frac{1}{2}}-4 \), we multiply the function \( \sqrt{3x} \) by - 1. When we multiply a non - negative number (the output of \( \sqrt{3x} \)) by - 1, the range of \( -(3x)^{\frac{1}{2}} \) becomes \( y\leq0 \) (because if \( a\geq0 \), then \( -a\leq0 \)).
Then we subtract 4 from \( -(3x)^{\frac{1}{2}} \). So for \( w(x)=-(3x)^{\frac{1}{2}}-4 \), if \( -(3x)^{\frac{1}{2}}\leq0 \), then \( w(x)=-(3x)^{\frac{1}{2}}-4\leq0 - 4=-4 \). So the range of \( w(x) \) is \( w(x)\leq - 4 \).

Answer:

domain: \( x\geq\boxed{0} \)
range: \( w(x)\leq\boxed{-4} \)