Choice D is correct. It’s given that the relationship between $x$ and $f (x)$ is linear. A linear function can be written in the form $f (x) = m x + b$ , where $m$ is the slope and $b$ is the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane. Given two points on a line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , the slope of the line can be found using the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . It’s given that the company’s profit is $dollar sign 220$ when it sells $8$ items and the profit is $dollar sign 320$ when it sells $10$ items. Since $f (x)$ represents the company's profit, in dollars, for selling $x$ items, the graph of $y = f (x)$ in the xy-plane passes through the points $left parenthesis 8 comma 220 right parenthesis$ and $left parenthesis 10 comma 320 right parenthesis$ . Substituting $left parenthesis 8 comma 220 right parenthesis$ and $left parenthesis 10 comma 320 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{320 - 220}{10 - 8}$ , which gives $m = \frac{100}{2}$ , or $m equals 50$ . Substituting $50$ for $m$ , $8$ for $x$ , and $220$ for $f (x)$ in $f (x) = m x + b$ yields $220 = (50) (8) + b$ , or $220 = 400 + b$ . Subtracting $400$ from each side of this equation yields $- 180 = b$ . Substituting $50$ for $m$ and $negative 180$ for $b$ in $f (x) = m x + b$ yields $f (x) = 50 x - 180$ . Therefore, the equation that defines $f$ is $f (x) = 50 x - 180$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.