The answer with the work has been provided previously.
An even function can be reflected over the y-axis and still remain unchanged.
Example: y=x^2
On the other hand, an odd function can be reflected around the origin and also remains unchanged.
Example: y=x^3
A straightforward method to determine this is:
if f(x) is even, then f(-x)=f(x)
if f(x) is odd, then f(-x)=-f(x)
Hence, for an even function
substitute -x in for each and check for equivalence
make sure to fully expand the expressions
g(x)=(x-1)^2+1=x^2-2x+1+1=x^2-2x+2 is the original expression
g(x)=(x-1)^2+1
g(-x)=(-x-1)^2+1
g(-x)=(1)(x+1)^2+1
g(-x)=x^2+2x+1+1
g(-x)=x^2+2x+2
Not the same, as the original contains -2x
Therefore, it is not even
g(x)=2x^2+1
g(-x)=2(-x)^2+1
g(-x)=2x^2+1
It matches, hence it is even
g(x)=4x+2
g(-x)=4(-x)+2
g(-x)=-4x+2
Not equivalent, thus not even
g(x)=2x
g(-x)=2(-x)
g(-x)=-2x
Not equal, therefore not even
g(x)=2x²+1 is the confirmed even function.
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