Answered

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Which statement verifies that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other?

A. [tex]\( f(g(x)) = x \)[/tex]

B. [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = -x \)[/tex]

C. [tex]\( f(g(x)) = \frac{1}{g(f(x))} \)[/tex]

D. [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]

Sagot :

GinnyAnswer
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]
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