The square root function \( k(x) = \sqrt{5x} \) can be related to a quadratic function by considering the inverse relationship between the two. Specifically, if \( y = \sqrt{5x} \), squaring both sides gives:
\[ y^2 = 5x \]
To express \( x \) in terms of \( y^2 \):
\[ x = \frac{y^2}{5} = \frac{1}{5}y^2 \]
Now, if we let \( m(y) = y^2 \), then the quadratic function can be expressed as:
\[ m(x) = \frac{1}{5}x^2 \]
However, if we're relating the square root function \( k(x) = \sqrt{5x} \) with a quadratic function \( m(x) \), we can understand that the coefficient in relationship to square \( k(x) = 5x \) suggests that all functions in response options are scaled multiples of \( x^2 \).
To find the specific function relating to the given square root function, we can multiply through the approximation \( x = \frac{1}{5}y^2 \).
None of the options given match exactly with this. The closest form, based on a quadratic function representation, that corresponds to the format involved here would possibly fall in the range of multiplying through the formulation.
Given the understanding of quadratic relationships, the response option that best aligns:
Option: \( m(x) = \frac{1}{25}x^2 \)
This choice caters well, ensuring an inverse proportionality to the square root derivation laid initially.