To determine whether [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other, we need to verify two conditions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
These conditions ensure that applying one function after the other returns the original input, which is the definition of inverse functions.
Here’s how we verify:
### Step-by-Step Verification
1. First Condition: [tex]\( f(g(x)) = x \)[/tex]
When you apply [tex]\( g(x) \)[/tex] to an input [tex]\( x \)[/tex] and then apply [tex]\( f(x) \)[/tex] to the result, you should get back the original input [tex]\( x \)[/tex]. Mathematically, this is written as:
[tex]\[
f(g(x)) = x
\][/tex]
This means that for every [tex]\( x \)[/tex] in the domain of [tex]\( g \)[/tex], [tex]\( f(g(x)) \)[/tex] returns [tex]\( x \)[/tex].
2. Second Condition: [tex]\( g(f(x)) = x \)[/tex]
Similarly, when you apply [tex]\( f(x) \)[/tex] to an input [tex]\( x \)[/tex] and then apply [tex]\( g(x) \)[/tex] to the result, you should also get back the original input [tex]\( x \)[/tex]. Mathematically, this is written as:
[tex]\[
g(f(x)) = x
\][/tex]
This means that for every [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex], [tex]\( g(f(x)) \)[/tex] returns [tex]\( x \)[/tex].
### Analysis of Given Statements
1. [tex]\( f(g(x)) = x \)[/tex]
This statement only verifies the first condition. It is necessary but not sufficient by itself for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to be inverses because it does not verify the second condition.
2. [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = -x \)[/tex]
This statement verifies the first condition correctly. However, the second condition is incorrect because it must be [tex]\( g(f(x)) = x \)[/tex], not [tex]\( g(f(x)) = -x \)[/tex].
3. [tex]\( f(g(x)) = \frac{1}{g(f(x))} \)[/tex]
Neither of these expressions aligns with the conditions required to verify inverse functions. Thus, this statement is incorrect.
4. [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex]
This statement verifies both conditions necessary to confirm that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are indeed inverses of each other.
Given these verifications, the correct statement that verifies [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are inverses of each other is:
[tex]\[
\boxed{4}
\][/tex]
