To determine if a function [tex]\( g(x) \)[/tex] is even, we need to check whether it satisfies the condition [tex]\( g(x) = g(-x) \)[/tex] for all [tex]\( x \)[/tex] in the domain of the function. Let's analyze each given function step by step:
1. [tex]\( g(x) = (x-1)^2 + 1 \)[/tex]
Let's substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(-x) = (-x - 1)^2 + 1
\][/tex]
Simplifying inside the parentheses:
[tex]\[
g(-x) = (-(x + 1))^2 + 1 = (x + 1)^2 + 1
\][/tex]
Therefore:
[tex]\[
(x+1)^2 = x^2 + 2x + 1
\][/tex]
Substituting back:
[tex]\[
g(-x) = x^2 + 2x + 1 + 1 = x^2 + 2x + 2 \neq x^2 - 2x + 2
\][/tex]
Since [tex]\( g(x) \neq g(-x) \)[/tex], this function is not even.
2. [tex]\( g(x) = 2x^2 + 1 \)[/tex]
Let's substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(-x) = 2(-x)^2 + 1
\][/tex]
Simplifying:
[tex]\[
g(-x) = 2x^2 + 1
\][/tex]
Since [tex]\( g(x) \)[/tex] also equals [tex]\( 2x^2 + 1 \)[/tex], we get:
[tex]\[
g(-x) = g(x)
\][/tex]
Therefore, this function is even.
3. [tex]\( g(x) = 4x + 2 \)[/tex]
Let's substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(-x) = 4(-x) + 2
\][/tex]
Simplifying:
[tex]\[
g(-x) = -4x + 2
\][/tex]
Since [tex]\( g(x) = 4x + 2 \)[/tex], we see:
[tex]\[
g(-x) \neq g(x)
\][/tex]
Therefore, this function is not even.
4. [tex]\( g(x) = 2x \)[/tex]
Let's substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
g(-x) = 2(-x)
\][/tex]
Simplifying:
[tex]\[
g(-x) = -2x
\][/tex]
Since [tex]\( g(x) = 2x \)[/tex], we see:
[tex]\[
g(-x) \neq g(x)
\][/tex]
Therefore, this function is not even.
Conclusively, among the provided functions, [tex]\( g(x) = 2x^2 + 1 \)[/tex] is the only even function.
