To determine the profit function [tex]\( P(x) \)[/tex], we need to subtract the cost function [tex]\( C(x) \)[/tex] from the revenue function [tex]\( R(x) \)[/tex]. Here are the given functions:
1. Revenue function: [tex]\( R(x) = 60x \)[/tex]
2. Cost function: [tex]\( C(x) = 12 + 7x \)[/tex]
The profit function [tex]\( P(x) \)[/tex] is defined as the difference between the revenue function and the cost function:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Let's perform the calculation step-by-step:
1. Write down the revenue function:
[tex]\[ R(x) = 60x \][/tex]
2. Write down the cost function:
[tex]\[ C(x) = 12 + 7x \][/tex]
3. Substitute [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the profit function formula:
[tex]\[ P(x) = 60x - (12 + 7x) \][/tex]
4. Distribute the negative sign through the parentheses:
[tex]\[ P(x) = 60x - 12 - 7x \][/tex]
5. Combine like terms:
[tex]\[ P(x) = (60x - 7x) - 12 \][/tex]
[tex]\[ P(x) = 53x - 12 \][/tex]
So, the profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 53x - 12 \][/tex]
