To determine the vertex of the quadratic function that represents the inverse of the given square root function \( g(x) \), we need to understand the relationship between a function and its inverse.
The general form of a square root function is:
\[ g(x) = \sqrt{x} \]
The inverse of this function can be found by switching the \( x \) and \( y \) in the equation:
\[ y = \sqrt{x} \implies x = \sqrt{y} \]
To find the inverse function, we can square both sides:
\[ x^2 = y \implies y = x^2 \]
The inverse of the square root function \( g(x) = \sqrt{x} \) is \( g^{-1}(x) = x^2 \).
The quadratic function \( y = x^2 \) is in its standard form where the vertex is located at the point (0, 0). Therefore, the vertex of the associated quadratic function is:
\[ (0, 0) \]
So the answer is:
(0, 0)