To find the inverse of a function at a specific point, we need to understand the relationship between the function [tex]\( f \)[/tex] and its inverse [tex]\( f^{-1} \)[/tex].
Given that [tex]\( f(-3) = 6 \)[/tex], this means that when we input [tex]\(-3\)[/tex] into the function [tex]\( f \)[/tex], the output is [tex]\( 6 \)[/tex]. The purpose of finding the inverse function [tex]\( f^{-1}(x) \)[/tex] is to determine what input into [tex]\( f \)[/tex] would yield a given output [tex]\( x \)[/tex].
In this case, since [tex]\( f(a) = b \)[/tex] implies [tex]\( f^{-1}(b) = a \)[/tex], we know that if [tex]\( f(-3) = 6 \)[/tex], then [tex]\( f^{-1}(6) \)[/tex] must be [tex]\(-3\)[/tex].
Therefore, [tex]\( f^{-1}(6) = -3 \)[/tex].
So, we fill in the blank as follows:
If a function [tex]\( f \)[/tex] has an inverse and [tex]\( f(-3)=6 \)[/tex], then [tex]\( f^{-1}(6) = \boxed{-3} \)[/tex].
