To find the profit function [tex]\( p(x) \)[/tex], we start by understanding the given functions:
1. Revenue Function: The revenue [tex]\( r(x) \)[/tex] from selling [tex]\( x \)[/tex] shirts is given by the equation:
[tex]\[
r(x) = 15x
\][/tex]
This means that each shirt is sold for [tex]$15.
2. Cost Function: The cost \( c(x) \) of buying \( x \) shirts is given by the equation:
\[
c(x) = 7x + 20
\]
This indicates that each shirt costs $[/tex]7 to buy, plus an additional fixed cost of $20, possibly representing overhead or initial cost.
3. Profit Function: The profit [tex]\( p(x) \)[/tex] is defined as the difference between the revenue and the cost. Therefore:
[tex]\[
p(x) = r(x) - c(x)
\][/tex]
Now, substitute the given functions into the profit equation:
[tex]\[
p(x) = (15x) - (7x + 20)
\][/tex]
Next, simplify the equation by distributing and combining like terms:
[tex]\[
p(x) = 15x - 7x - 20
\][/tex]
[tex]\[
p(x) = 8x - 20
\][/tex]
Therefore, the profit function [tex]\( p(x) \)[/tex] is:
[tex]\[
\boxed{8x - 20}
\][/tex]
