To determine which statement verifies that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other, let's go through each option step by step:
1. Option 1: [tex]\( f(g(x)) = x \)[/tex]
This condition alone is necessary but not sufficient to verify that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other. For [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to be true inverses, we need to check the relationship in both directions, not just one.
2. Option 2: [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = -x \)[/tex]
This condition cannot be true for demonstrating that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses. The relationship shows inverse behavior only for [tex]\( f(g(x)) \)[/tex], but changing the sign in [tex]\( g(f(x)) \)[/tex] to [tex]\( -x \)[/tex] disrupts the symmetry required for inverse functions. Inverse functions must map back to the original input, so this cannot be the correct option.
3. Option 3: [tex]\( f(g(x)) = \frac{1}{g(f(x))} \)[/tex]
This condition does not define the inverse relationship. The compositional output [tex]\( f(g(x)) \)[/tex] must return the input [tex]\( x \)[/tex] directly, not a fraction or any other form. This condition likely represents a different functional relationship, but not that of inverse functions.
4. Option 4: [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]
This condition is both necessary and sufficient to verify that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other. For [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to be true inverses, composing them in either order [tex]\( g(f(x)) \)[/tex] or [tex]\( f(g(x)) \)[/tex] should return the original input [tex]\( x \)[/tex]. This means we must have both:
[tex]\[
f(g(x)) = x
\][/tex]
and
[tex]\[
g(f(x)) = x
\][/tex]
Hence, the correct statement that verifies [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other is:
[tex]\[ f(g(x)) = x \text{ and } g(f(x)) = x \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{4} \][/tex]
