Question

Find the related square root function, p(x) for the quadratic equation h(x) = 4x ^ 2 with the domain of (0, ∞) (1 point) p(x) = sqrt(ax) where a =

Answer 1

To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \), we need to express this in terms of a square root function.

Given that \( h(x) = 4x^2 \), we can express \( x \) in terms of \( h(x) \):

\[
h(x) = 4x^2
\]

To find the square root, we take:

\[
x^2 = \frac{h(x)}{4}
\]

Taking the square root of both sides gives:

\[
x = \sqrt{\frac{h(x)}{4}} = \frac{1}{2} \sqrt{h(x)}
\]

Rearranging gives us:

\[
h(x) = 4x^2 \implies x = \frac{1}{2} \sqrt{h(x)}
\]

To formulate this in terms of \( p(x) \) where \( p(x) = \sqrt{ax} \), we can derive \( p(x) \) for the function where \( x \) is a function of \( h \). Starting with \( h(x) \):

Setting \( p(x) = \sqrt{ax} \) means we want to find \( a \) such that \( a \) relates our original quadratic function. We need to find \( p(h^{-1}(x)) \):

1. Rearranging \( h(x)= 4x^2 \) gives:

\[
x = \frac{1}{2} \sqrt{h(x)}
\]

2. Therefore, we can express \( h(x) \) in terms of \( x \) using:

\[
p(x) = \sqrt{4x}
\]

which can be expressed as

\[
p(x) = \sqrt{4} \sqrt{x} = 2\sqrt{x}
\]

Thus, for the form \( p(x) = \sqrt{ax} \), we identify \( a \) as:

\[
\sqrt{a} = 2 \implies a = 4
\]

Thus, the value of \( a \) is

\[
\boxed{4}
\]