\begin{tabular}{l} Determine the value of \( \mathfrak{p}^{b} \) and \( q \) if \( : y=b^{x+1}+q \) \\ \( \qquad \) \\ \hline\( (-3 ; 12) ; A \)\end{tabular}
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To determine the values of \( \mathfrak{p}^b \) and \( q \) in the equation \( y = b^{x+1} + q \) given the point \( (-3, 12) \), we start by substituting \( x = -3 \) and \( y = 12 \) into the equation: \[ 12 = b^{-3+1} + q \] This simplifies to: \[ 12 = b^{-2} + q \] Now, we know \( b^{-2} = \frac{1}{b^2} \). Thus, we can rewrite the equation as: \[ 12 = \frac{1}{b^2} + q \] From here, we want to express \( q \) in terms of \( b \): \[ q = 12 - \frac{1}{b^2} \] Now, we also need a value for \( b \) to find \( \mathfrak{p}^b \). But without an additional point or information, we can't calculate specific numerical values for \( \mathfrak{p}^b \) and \( q \). We can express everything in terms of \( b \) and \( q \) as: - For any chosen \( b \), we can find \( q \) using \( q = 12 - \frac{1}{b^2} \). - For example, if we let \( b = 2 \), then: \[ q = 12 - \frac{1}{2^2} = 12 - \frac{1}{4} = 12 - 0.25 = 11.75 \] So, \( q \) will change based on the value of \( b \), and we can determine \( \mathfrak{p}^b \) based on a chosen \( b \) as well. If you can provide another point or constraint, we can zero in on specific values!
