Let's analyze each option to determine which one must be an even function, given that [tex]\( g(x) \)[/tex] is an odd function. Remember that a function [tex]\( g(x) \)[/tex] is odd if [tex]\( g(-x) = -g(x) \)[/tex], and a function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex].
### Option 1: [tex]\( f(x) = g(x) + 2 \)[/tex]
To check if [tex]\( f(x) = g(x) + 2 \)[/tex] is an even function, we need to evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = g(-x) + 2
\][/tex]
Since [tex]\( g(x) \)[/tex] is odd, [tex]\( g(-x) = -g(x) \)[/tex]. Thus,
[tex]\[
f(-x) = -g(x) + 2
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = g(x) + 2 \quad \text{and} \quad f(-x) = -g(x) + 2
\][/tex]
Since [tex]\( f(-x) \ne f(x) \)[/tex], [tex]\( f(x) = g(x) + 2 \)[/tex] is not an even function.
### Option 2: [tex]\( f(x) = g(x) + g(x) \)[/tex]
To check if [tex]\( f(x) = g(x) + g(x) \)[/tex] is an even function:
[tex]\[
f(x) = 2g(x)
\][/tex]
We need to evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 2g(-x) = 2(-g(x)) = -2g(x)
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 2g(x) \quad \text{and} \quad f(-x) = -2g(x)
\][/tex]
Since [tex]\( f(-x) \ne f(x) \)[/tex], [tex]\( f(x) = g(x) + g(x) \)[/tex] is not an even function.
### Option 3: [tex]\( f(x) = g(x)^2 \)[/tex]
To check if [tex]\( f(x) = g(x)^2 \)[/tex] is an even function:
[tex]\[
f(x) = g(x)^2
\][/tex]
We need to evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (g(-x))^2
\][/tex]
Since [tex]\( g(x) \)[/tex] is odd, [tex]\( g(-x) = -g(x) \)[/tex]. Thus,
[tex]\[
f(-x) = (-g(x))^2 = g(x)^2
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = g(x)^2 \quad \text{and} \quad f(-x) = g(x)^2
\][/tex]
Since [tex]\( f(-x) = f(x) \)[/tex], [tex]\( f(x) = g(x)^2 \)[/tex] is indeed an even function.
### Option 4: [tex]\( f(x) = -g(x) \)[/tex]
To check if [tex]\( f(x) = -g(x) \)[/tex] is an even function:
[tex]\[
f(x) = -g(x)
\][/tex]
We need to evaluate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = -g(-x)
\][/tex]
Since [tex]\( g(x) \)[/tex] is odd, [tex]\( g(-x) = -g(x) \)[/tex]. Thus,
[tex]\[
f(-x) = -(-g(x)) = g(x)
\][/tex]
Comparing [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = -g(x) \quad \text{and} \quad f(-x) = g(x)
\][/tex]
Since [tex]\( f(-x) \ne f(x) \)[/tex], [tex]\( f(x) = -g(x) \)[/tex] is not an even function.
### Conclusion
The function that must be an even function is [tex]\( f(x) = g(x)^2 \)[/tex]. Therefore, the correct option is:
[tex]\[
\boxed{3}
\][/tex]
