To find 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].
Given:
[tex]\[ r(x) = 15x \][/tex]
[tex]\[ c(x) = 7x + 20 \][/tex]
The profit function [tex]\( p(x) \)[/tex] is calculated as follows:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
Substitute the given functions for [tex]\( r(x) \)[/tex] and [tex]\( c(x) \)[/tex]:
[tex]\[ p(x) = 15x - (7x + 20) \][/tex]
Distribute the negative sign:
[tex]\[ p(x) = 15x - 7x - 20 \][/tex]
Combine like terms:
[tex]\[ p(x) = (15x - 7x) - 20 \][/tex]
[tex]\[ p(x) = 8x - 20 \][/tex]
Thus, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[ p(x) = 8x - 20 \][/tex]
Therefore, the correct answer is:
D. [tex]\( p(x) = 8x - 20 \)[/tex]
