To determine the profit function [tex]\((k - h)(x)\)[/tex], we need to subtract the cost function [tex]\(h(x)\)[/tex] from the revenue function [tex]\(k(x)\)[/tex].
Let's start by writing down the given functions:
- Cost function: [tex]\(h(x) = 5x + 6\)[/tex]
- Revenue function: [tex]\(k(x) = 9x - 2\)[/tex]
The profit function is:
[tex]\[
(k - h)(x) = k(x) - h(x)
\][/tex]
Substitute the given expressions for [tex]\(k(x)\)[/tex] and [tex]\(h(x)\)[/tex]:
[tex]\[
(k - h)(x) = (9x - 2) - (5x + 6)
\][/tex]
Next, we need to simplify this expression:
First, distribute the negative sign:
[tex]\[
(9x - 2) - (5x + 6) = 9x - 2 - 5x - 6
\][/tex]
Combine like terms by grouping the [tex]\(x\)[/tex] terms and the constant terms separately:
Group the [tex]\(x\)[/tex] terms:
[tex]\[
9x - 5x = 4x
\][/tex]
Group the constant terms:
[tex]\[
-2 - 6 = -8
\][/tex]
Now combine these results to get the simplified expression:
[tex]\[
4x - 8
\][/tex]
Therefore, the expression that represents the profit [tex]\((k - h)(x)\)[/tex] of producing soccer balls is:
[tex]\[
4x - 8
\][/tex]
Among the given options, this matches option:
[tex]\[
\boxed{4x - 8}
\][/tex]
