Sure, let's determine the profit function [tex]\( P(x) \)[/tex] step by step.
1. Define the Revenue Function:
The revenue function represents the total income generated from selling [tex]\( x \)[/tex] video games. This function is given as:
[tex]\[ R(x) = 60x \][/tex]
Here, 60 represents the price at which each video game is sold.
2. Define the Cost Function:
The cost function represents the total cost incurred in producing and selling [tex]\( x \)[/tex] video games. This is given as:
[tex]\[ C(x) = 12 + 7x \][/tex]
Here, 12 represents the fixed costs (costs that do not change with the number of video games produced), and 7x represents the variable costs (costs that vary with the number of video games produced).
3. Define the Profit Function:
The profit function is determined by subtracting the cost function from the revenue function. This represents the net profit made from selling [tex]\( x \)[/tex] video games.
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substituting the given revenue and cost functions into this formula, we get:
[tex]\[ P(x) = 60x - (12 + 7x) \][/tex]
4. Simplify the Profit Function:
To simplify the profit function, we need to distribute the negative sign through the cost function and combine like terms:
[tex]\[
\begin{align*}
P(x) &= 60x - 12 - 7x \\
P(x) &= (60x - 7x) - 12 \\
P(x) &= 53x - 12
\end{align*}
\][/tex]
Therefore, the profit function is:
[tex]\[ P(x) = 53x - 12 \][/tex]
This means the profit from selling [tex]\( x \)[/tex] video games is represented by the equation [tex]\( P(x) = 53x - 12 \)[/tex], where 53 is the net profit per video game after accounting for costs, and 12 is the fixed cost that must be deducted from the total profit.
