To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with the domain \( (0, \infty) \), we can follow these steps:
-
Start with the equation: \[ h(x) = 4x^2 \]
-
To express this in terms of a square root function, we can isolate \( x \): To find \( x \) in terms of \( h(x) \), we can take the square root of both sides. But first, let’s rewrite the equation in a usable form: \[ h(x) = y \quad \text{(where \( y \) is the output of the function)} \] This gives us: \[ y = 4x^2 \]
-
Solving for \( x \): To express \( x \), we rearrange this as follows: \[ x^2 = \frac{y}{4} \] Now, taking the square root of both sides: \[ x = \sqrt{\frac{y}{4}} = \frac{\sqrt{y}}{2} \]
-
Relating back to \( p(x) \): Since \( p(x) \) will be defined in terms of \( y \), we can replace \( y \) with \( p(x) \) to give the inverse relation. Thus: \[ p(y) = \frac{\sqrt{y}}{2} \]
-
Final form: Therefore, the related square root function \( p(y) \) to the quadratic function \( h(x) = 4x^2 \) is: \[ p(y) = \frac{\sqrt{y}}{2} \]
In summary, the related square root function \( p(y) \) for the quadratic equation \( h(x) = 4x^2 \) defined over the domain \( (0, \infty) \) is: \[ p(y) = \frac{\sqrt{y}}{2} \]