Answer: Second Option
[tex]g(x) = 2x^2 + 1[/tex]
Step-by-step explanation:
By definition, a function f(x) is an even function if:
[tex]f (-x) = f (x)[/tex]
This means that each input value x and its negative -x are assigned the same output value y.
To verify which of the functions is even, you must test [tex]f(-x) = f(x)[/tex] for each of them
First option
[tex]g(x) = (x - 1)^2 + 1[/tex]
[tex]g(-x) = (-x -1)^2 +1\\\\g(-x) = ((-1)(x+1))^2 +1\\\\g(-x) = (-1)^2(x+1)^2 +1\\\\g(-x) = (x+1)^2 +1\neq g(x)[/tex]
Second option
[tex]g(x) = 2x^2 + 1[/tex]
[tex]g(-x) = 2(-x)^2 + 1[/tex]
[tex]g(-x) = 2x^2 + 1=g(x)[/tex]
Third option
[tex]g(x) = 4x + 2[/tex]
[tex]g(-x) = 4(-x) + 2[/tex]
[tex]g(-x) = -4x + 2\neq g(x)[/tex]
Fourth option
[tex]g(x) = 2^x[/tex]
[tex]g(-x) = 2^(-x)[/tex]
[tex]g(-x) = \frac{1}{2^x}\neq g(x)[/tex]
