Part A: To write the exponential function that represents the value of the car at the end of every year, we start with the initial value of the car which is $40,000. The car depreciates by 15% each year, meaning it retains 85% of its value. This can be written as the equation:
\[ f(x) = 40000(0.85)^x \]
Part B: Now we will fill out the table for the value of the car after x years (0, 1, 3, 5, 8):
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For \( x = 0 \): \[ f(0) = 40000(0.85)^0 = 40000(1) = 40000.00 \]
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For \( x = 1 \): \[ f(1) = 40000(0.85)^1 = 40000(0.85) = 34000.00 \]
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For \( x = 3 \): \[ f(3) = 40000(0.85)^3 = 40000(0.614125) \approx 24565.00 \]
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For \( x = 5 \): \[ f(5) = 40000(0.85)^5 = 40000(0.443705) \approx 17748.20 \]
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For \( x = 8 \): \[ f(8) = 40000(0.85)^8 = 40000(0.272490) \approx 10899.60 \]
So the table representing the value of the car after x years is as follows:
\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline 0 & 40000.00 \ 1 & 34000.00 \ 3 & 24565.00 \ 5 & 17748.20 \ 8 & 10899.60 \ \hline \end{array} \]