To find the quadratic function related to the square root function \( k(x) = 5\sqrt{x} \), we can express this in terms of the relationship between the square root function and its corresponding quadratic function.
Given the square root function, if we isolate \( x \), we can think of it as a square function in the following way:
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Start with the function: \[ k(x) = 5\sqrt{x} \]
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To find the corresponding quadratic function, we can square both sides: \[ k(x)^2 = (5\sqrt{x})^2 \] \[ m(x) = 25x \]
However, none of the options presented corresponds directly to the expression \( m(x) = 25x \).
If we consider possible transformations without isolating \( k(x) \), we can recognize that one of the options must represent a transformation of the basic relationship \( y = x^2 \).
So, we will take a look again at the relationship:
- If \( k(x) = 5\sqrt{x} \) corresponds to squaring and yielding a parabola, then we'd consider the scaling factor due to the square root.
To summarize: \[ k(x) = 5\sqrt{x} \implies m(x) = 25x \]
From your choices, let's analyze them:
- \( m(x) = \frac{1}{5}x^2 \) (not suitable)
- \( m(x) = 5x^2 \) (also a quadratic, but does not relate directly)
- \( m(x) = \frac{1}{25}x^2 \) (not suitable)
- \( m(x) = 25x^2 \) (matches the form we derived).
Given the above analysis, the best match for a quadratic function corresponding to the square root function \( k(x) = 5\sqrt{x} \) is:
\( m(x) = 25x \). However, please check if the question might have been misinterpreted, since the form directly derived does not seem available. If you're looking for a transformation-based quadratic, \( m(x) = 25x^2 \) would be a conceivable evaluation for parabolic growth from our original square root function, although it oversimplifies direct correlation.
Please review to confirm the intentions behind options or context!