To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with the domain \( (0, \infty) \), we can express \( h(x) \) in terms of \( p(x) \).
Given: \[ h(x) = 4x^2 \]
To derive the square root function \( p(x) \), we start by solving for \( x \) in terms of \( h(x) \):
- Set \( y = h(x) \) so that \( y = 4x^2 \).
- To isolate \( x \), divide both sides by 4: \[ \frac{y}{4} = x^2 \]
- Now take the square root of both sides. Since we are considering the domain \( (0, \infty) \), we only take the positive root: \[ x = \sqrt{\frac{y}{4}} = \frac{1}{2} \sqrt{y} \]
This implies that: \[ p(y) = \frac{1}{2} \sqrt{y} \]
Thus, the related square root function \( p(x) \), as a function of \( h(x) \), is: \[ p(x) = \frac{1}{2} \sqrt{x} \]
In summary, the related square root function for the quadratic equation \( h(x) = 4x^2 \) is: \[ p(x) = \frac{1}{2} \sqrt{x} \] with the domain \( (0, \infty) \).