To determine which function is even, we need to understand the definition of an even function. A function [tex]\( f(x) \)[/tex] is classified as even if it satisfies the condition [tex]\( f(x) = f(-x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's analyze each given function step by step:
1. Function [tex]\( f(x) = x^3 + 2 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function:
[tex]\[
f(-x) = (-x)^3 + 2 = -x^3 + 2
\][/tex]
- Check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
x^3 + 2 \neq -x^3 + 2
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], this function is not even.
2. Function [tex]\( f(x) = x^2 + 1 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function:
[tex]\[
f(-x) = (-x)^2 + 1 = x^2 + 1
\][/tex]
- Check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
x^2 + 1 = x^2 + 1
\][/tex]
- Since [tex]\( f(x) = f(-x) \)[/tex], this function is even.
3. Function [tex]\( f(x) = |x + 2| \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function:
[tex]\[
f(-x) = |-x + 2| = |-(x - 2)| = |x - 2|
\][/tex]
- Check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
|x + 2| \neq |x - 2|
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], this function is not even.
4. Function [tex]\( f(x) = \sin(2x) \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex] in the function:
[tex]\[
f(-x) = \sin(2(-x)) = \sin(-2x)
\][/tex]
- Recall that [tex]\( \sin(-\theta) = -\sin(\theta) \)[/tex]:
[tex]\[
\sin(-2x) = -\sin(2x)
\][/tex]
- Check if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
\sin(2x) \neq -\sin(2x)
\][/tex]
- Since [tex]\( f(x) \neq f(-x) \)[/tex], this function is not even.
Based on the analysis, the function that is even is:
[tex]\[
\boxed{2}
\][/tex]
