To find Luisa's profit function, we need to follow the relationship given by [tex]\( p(x) = r(x) - c(x) \)[/tex], where [tex]\( r(x) \)[/tex] is the revenue function and [tex]\( c(x) \)[/tex] is the cost function.
1. Start by writing down the revenue function:
[tex]\[ r(x) = 20x \][/tex]
2. Write down the cost function:
[tex]\[ c(x) = 4x + 25 \][/tex]
3. To find the profit function [tex]\( p(x) \)[/tex], subtract the cost function from the revenue function:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
4. Substitute the given functions for [tex]\( r(x) \)[/tex] and [tex]\( c(x) \)[/tex] into the equation:
[tex]\[ p(x) = 20x - (4x + 25) \][/tex]
5. Distribute the negative sign across the terms in the cost function:
[tex]\[ p(x) = 20x - 4x - 25 \][/tex]
6. Combine like terms:
[tex]\[ p(x) = (20x - 4x) - 25 \][/tex]
[tex]\[ p(x) = 16x - 25 \][/tex]
Therefore, the profit function [tex]\( p(x) = 16x - 25 \)[/tex], which corresponds to option D.
