To determine the profit function, we start by understanding the given cost and revenue functions.
The cost function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[ h(x) = 5x + 6 \][/tex]
The revenue function [tex]\( k(x) \)[/tex] is defined as:
[tex]\[ k(x) = 9x - 2 \][/tex]
The profit function [tex]\( (k - h)(x) \)[/tex] is derived by subtracting the cost function from the revenue function:
[tex]\[ (k - h)(x) = k(x) - h(x) \][/tex]
Now, substitute the expressions for [tex]\( k(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[ (k - h)(x) = (9x - 2) - (5x + 6) \][/tex]
Next, distribute the subtraction:
[tex]\[ (k - h)(x) = 9x - 2 - 5x - 6 \][/tex]
Combine like terms:
[tex]\[ (k - h)(x) = 9x - 5x - 2 - 6 \][/tex]
[tex]\[ (k - h)(x) = 4x - 8 \][/tex]
Thus, the expression that represents the profit function is:
[tex]\[ 4x - 8 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4x - 8} \][/tex]
