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