To determine the correct choices that complete the sentence, let's first understand the properties of inverse functions. If two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other, this means that when [tex]\( f \)[/tex] is applied to the output of [tex]\( g \)[/tex], or [tex]\( g \)[/tex] is applied to the output of [tex]\( f \)[/tex], the result is the original input [tex]\( x \)[/tex]. Formally, this relationship is expressed by:
[tex]\[ (f \circ g)(x) = f(g(x)) = x \][/tex]
[tex]\[ (g \circ f)(x) = g(f(x)) = x \][/tex]
Given the sentence:
"If two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses, then [tex]\( (f \circ g)(x) = g(x) \)[/tex] and [tex]\(\square\)[/tex] [tex]\( g(x) = x\)[/tex]."
To match the second property stated above, the correct choice to fill in the blank is:
[tex]\[ (g \circ f)(x) \][/tex]
Thus, the completed sentence reads:
"If two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses, then [tex]\( (f \circ g)(x) = g(x) \)[/tex] and [tex]\( (g \circ f)(x) = x \)[/tex]."
