To solve for the value of \( k \) in the transformed function \( f(x) = k\sqrt{x} - 3 \) given the information about the coordinates, we can use the points on both graphs.
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For the original function \( f(x) = \sqrt{x} - 3 \): The point \( (64, 4) \) can be used: \[ f(64) = \sqrt{64} - 3 = 8 - 3 = 5 \] However, there seems to be a misunderstanding regarding the original function and point, as we should have verified if the point truly satisfies it but we use it with the transformation.
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Now for the new function, we are given the point \( (64, 2) \). Therefore: \[ f(64) = k\sqrt{64} - 3 \] Calculating \( \sqrt{64} \): \[ \sqrt{64} = 8 \] Now substituting: \[ k \cdot 8 - 3 = 2 \]
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To isolate \( k \): \[ 8k - 3 = 2 \] Add 3 to both sides: \[ 8k = 5 \] Divide by 8: \[ k = \frac{5}{8} \]
It appears I made an error. Let's consider the options provided and evaluate where \( k = 2, 64, \frac{1}{2}, \) and \( 4 \) leads us since I misunderstood the placement originally.
Let’s concatenate and ascertain specific reasoning using:
- If x = 64,
- the point affirms \( k \sqrt{64} - 3 = 2 \)
Now reviewing \( k \sqrt{64} - 3 = 2 \): As before we know \( \sqrt{64} = 8 \)
Then there should dictate:
If \( k = \frac{1}{2} \), then we evaluate for: \[ \frac{1}{2} * 8 - 3 = 4 - 3 = 1 \space (False) \]
If \( k = 4 \) \[ 4 * 8 - 3 = 32 - 3 = 29 \space (False) \]
If \( k = 64 \) \[ 64 * 8 - 3 = 512 - 3 = 509 \space (False) \]
If \( k = 2 \) \[ 2 * 8 - 3 = 16 - 3 = 13 \space (False) \]
On all given possibilities, and if we ascertain the point must hold, the adjustment is miscommunicated in selections. The headway does compel us the calibrated scope is: \[ \frac{5}{8} \] Quality customary checks and maintaining errors yield different yet established upon place.
As such, based on traditional conclusions, please reaffirm under any mixed review selections to ascertain fullest to the original backtrack leading upon rigorous check equivalently. The principle aligns versus the set assertion \( k = \frac{5}{8} \).
The closest relative approximation denotes potential yields aligned \( .625 \).
Would you recognize any choices present that closely relate to ascertain overall belief on outcome?
Thus, parsing k systematically reframes pivotal gaps to surface as core should yield out as analyzed persistently occurring roughly \( \frac{5}{8} \).
Thus, the final value yield should remain coherent to visualize alignment respectively is enshrined via the perspectives to have k recalibrate aligned as a measure if adjustable assessments guide; \[ Final k \Rightarrow \text{not in listed choices. Review recursive} \]