Question

Find the related square root function, for the quadratic equation h(x)=4x2
with the domain of (0,∞)
p(x)=ax−−√
where a=

Answer 1

To find the related square root function for the quadratic equation \( h(x) = 4x^2 \) with a domain of \( (0, \infty) \), we first recognize that a square root function can be derived from the quadratic function by expressing \( x \) in terms of \( h(x) \).

1. Start from the equation:
\[
h(x) = 4x^2
\]

2. To express \( x \) in terms of \( h(x) \), solve for \( x \):
\[
x^2 = \frac{h(x)}{4}
\]
\[
x = \sqrt{\frac{h(x)}{4}} = \frac{\sqrt{h(x)}}{2}
\]

3. Now, to write it in terms of \( x \) as you defined \( p(x) = a\sqrt{x} \), notice that the equation \( h(x) = 4x^2 \) implies \( \sqrt{h(x)} = 2x \).

Therefore, if we take \( h(x) \) as previously established, we want to relate it to the square root form.

Given the form \( p(x) \), if we set:
\[
p(x) = 2\sqrt{x}
\]

We can define \( a \) as \( a = 2 \).

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

Hence, the value of \( a \) is:

\[
\boxed{2}
\]