Question: Indicate which of the following four graphs is the correct graph of this function. \[ p(x) =…

Indicate which of the following four graphs is the correct graph of this function.

\[ p(x) = \frac{1}{\left(\frac{3}{2}\right)^x} - 3 \]

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The function given is \[ p(x) = \left( \frac{1}{\frac{3}{2}} \right)^x - 3 \] First, simplify the expression: \[ \frac{1}{\frac{3}{2}} = \frac{2}{3} \] Thus, the function simplifies to: \[ p(x) = \left( \frac{2}{3} \right)^x - 3 \] This is an exponential decay function because the base \(\frac{2}{3}\) is between 0 and 1. The graph of an exponential decay function like this decreases as \(x\) increases. Additionally, the entire graph is shifted downward by 3 units because of the \(-3\) at the end of the equation. Look for the graph that starts above \(y = -3\) and decreases, approaching \(y = -3\) as \(x\) increases, but never reaching it. Based on the given options, the graph in the second position from the left in the top row is the correct one.