Question

which of the following is an even function?
○ g(x) = (x − 1)² + 1
○ g(x) = 2x² + 1
○ g(x) = 4x + 2
○ g(x) = 2x

Explanation:

Step1: Recall the definition of an even function

A function \( g(x) \) is even if \( g(-x)=g(x) \) for all \( x \) in the domain of \( g \).

Step2: Test \( g(x)=(x - 1)^2+1 \)

First, find \( g(-x) \):
\( g(-x)=(-x - 1)^2+1=(x + 1)^2+1 \)
Compare with \( g(x)=(x - 1)^2+1 \). Since \( (x + 1)^2+1
eq(x - 1)^2+1 \) (for example, when \( x = 1 \), \( g(-1)=( - 1+1)^2+1 = 1 \), \( g(1)=(1 - 1)^2+1 = 1 \); wait, no, let's expand \( (x - 1)^2+1=x^{2}-2x + 1+1=x^{2}-2x + 2 \), and \( g(-x)=(-x - 1)^2+1=(x + 1)^2+1=x^{2}+2x + 2 \). So \( g(-x)
eq g(x) \), so not even.

Step3: Test \( g(x)=2x^{2}+1 \)

Find \( g(-x) \):
\( g(-x)=2(-x)^{2}+1=2x^{2}+1 \)
Since \( g(-x)=g(x) \), this function is even.

Step4: Test \( g(x)=4x + 2 \)

Find \( g(-x) \):
\( g(-x)=4(-x)+2=-4x + 2 \)
Compare with \( g(x)=4x + 2 \). \( -4x + 2
eq4x + 2 \) (unless \( x = 0 \)), so not even.

Step5: Test \( g(x)=2x \)

Find \( g(-x) \):
\( g(-x)=2(-x)=-2x=-g(x) \)
This is an odd function, not even.

Answer:

\( g(x) = 2x^2 + 1 \) (the second option)