Answer :
To find the value of the inverse function [tex]\( f^{-1}(6) \)[/tex], we need to understand the relationship between a function and its inverse.
Given: [tex]\( f(-3) = 6 \)[/tex].
To understand this in the context of the inverse function, we should know that the inverse function, [tex]\( f^{-1} \)[/tex], essentially "reverses" what the original function [tex]\( f \)[/tex] does. Therefore, if [tex]\( f(x) \)[/tex] maps [tex]\( x \)[/tex] to [tex]\( y \)[/tex], then [tex]\( f^{-1}(y) \)[/tex] will map [tex]\( y \)[/tex] back to [tex]\( x \)[/tex].
In this case, [tex]\( f(-3) = 6 \)[/tex] means that when the input to the function [tex]\( f \)[/tex] is [tex]\(-3\)[/tex], the output is [tex]\( 6 \)[/tex].
So, for the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(6) \][/tex]
should give us the input value [tex]\(-3\)[/tex], because the inverse function undoes the operation of the original function.
Therefore, if [tex]\( f(-3) = 6 \)[/tex], then:
[tex]\[ f^{-1}(6) = -3 \][/tex]
Hence, the value that fills in the blank is [tex]\(-3\)[/tex].
So, completing the statement:
If a function [tex]\( f \)[/tex] has an inverse and [tex]\( f(-3) = 6 \)[/tex], then [tex]\( f^{-1}(6) = -3 \)[/tex].
Given: [tex]\( f(-3) = 6 \)[/tex].
To understand this in the context of the inverse function, we should know that the inverse function, [tex]\( f^{-1} \)[/tex], essentially "reverses" what the original function [tex]\( f \)[/tex] does. Therefore, if [tex]\( f(x) \)[/tex] maps [tex]\( x \)[/tex] to [tex]\( y \)[/tex], then [tex]\( f^{-1}(y) \)[/tex] will map [tex]\( y \)[/tex] back to [tex]\( x \)[/tex].
In this case, [tex]\( f(-3) = 6 \)[/tex] means that when the input to the function [tex]\( f \)[/tex] is [tex]\(-3\)[/tex], the output is [tex]\( 6 \)[/tex].
So, for the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(6) \][/tex]
should give us the input value [tex]\(-3\)[/tex], because the inverse function undoes the operation of the original function.
Therefore, if [tex]\( f(-3) = 6 \)[/tex], then:
[tex]\[ f^{-1}(6) = -3 \][/tex]
Hence, the value that fills in the blank is [tex]\(-3\)[/tex].
So, completing the statement:
If a function [tex]\( f \)[/tex] has an inverse and [tex]\( f(-3) = 6 \)[/tex], then [tex]\( f^{-1}(6) = -3 \)[/tex].
