4Question: A chemist models the efficiency of a new green catalyst with the function $ f(x) = x^2 - 6x + k $, where $ x $ represents reaction temperature and $ k $ is a parameter related to material properties. If the catalyst achieves peak efficiency when $ x = 3 $, what is the value of $ k $ such that $ f(3) = 0 $?

To determine the value of $ k $ that ensures peak efficiency at $ x = 3 $, we begin by analyzing the given function:

$$f(x) = x^2 - 6x + k$$

Peak efficiency occurs at $ x = 3 $, and we are told that $ f(3) = 0 $. This implies that $ x = 3 $ is a root of the function. Substituting $ x = 3 $ into the equation:

$$f(3) = (3)^2 - 6(3) + k = 0$$

Simplify the expression:

$$9 - 18 + k = 0$$

$$-9 + k = 0$$

$$k = 9$$

Thus, the value of $ k $ that ensures peak efficiency at $ x = 3 $ is:

$$\boxed{9}$$