To determine which rule could not be a function for \( f(x) \) given that \( f(2) = 7 \), we need to evaluate each of the provided function rules at \( x = 2 \).
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For \( f(x) = |2x - 5| + 6 \): \[ f(2) = |2(2) - 5| + 6 = |4 - 5| + 6 = | - 1 | + 6 = 1 + 6 = 7 \] This function satisfies \( f(2) = 7 \).
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For \( f(x) = 2x + 3 \): \[ f(2) = 2(2) + 3 = 4 + 3 = 7 \] This function also satisfies \( f(2) = 7 \).
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For \( f(x) = 2[x] + 3 \): Here, \( [x] \) denotes the greatest integer function (floor function). Thus, for \( x = 2 \): \[ f(2) = 2[2] + 3 = 2(2) + 3 = 4 + 3 = 7 \] This function satisfies \( f(2) = 7 \) as well.
Since all provided function rules satisfy \( f(2) = 7 \), based on the information and evaluations given, all of them could technically be function rules for \( f(x) \). However, if you are looking to identify a scenario or a different constraint that might limit one of these functions, more context would be needed.
If you are specifically asked which one might be less typical (or least likely) based on context, we can mention that functions involving floor values (like \( f(x) = 2[x] + 3 \)) can be less straightforward, especially in understanding continuity or outputs for non-integer inputs, but in terms of the specific output at \( x = 2 \), all functions provided are valid.
Thus, none of these functions could "not" be a function rule for \( f(x) \) based on the specific evaluation at \( x = 2 \). All of them validly produce \( f(2) = 7 \).