To solve the problem, let’s first understand the function [tex]\( f(x) = -x^2 + 18x - 72 \)[/tex]. This function represents the daily profit in dollars depending on the number of monthly memberships sold, where [tex]\( x \)[/tex] is the number of memberships.
The zeros of the function are the values of [tex]\( x \)[/tex] where the profit, [tex]\( f(x) \)[/tex], is equal to zero. In other words, we need to solve the equation:
[tex]\[ f(x) = -x^2 + 18x - 72 = 0 \][/tex]
By solving this quadratic equation, we find the values of [tex]\( x \)[/tex] that make the profit zero. These values of [tex]\( x \)[/tex] are known as the zeros of the function.
The zeros for this function are:
[tex]\[ x = 6 \][/tex]
[tex]\[ x = 12 \][/tex]
This means when [tex]\( 6 \)[/tex] or [tex]\( 12 \)[/tex] memberships are sold, the gym makes no profit ([tex]\( f(x) = 0 \)[/tex]).
So, in the context of the problem, the zeros [tex]\( x = 6 \)[/tex] and [tex]\( x = 12 \)[/tex] represent the number of monthly memberships when no profit is made.
Therefore, the correct interpretation is:
[tex]\[ x = 6, x = 12 \][/tex]
The zeros represent the number of monthly memberships when no profit is made.
