Question 996033
Claim: g(x) is an even function


Proof:

We need to show that g(x) = g(-x) is true for any real number x.


g(x) = -2[f(x)+f(-x)]
g(-x) = -2[f(-x)+f(-(-x))] ... replace EVERY x with -x
g(-x) = -2[f(-x)+f(x)]
g(-x) = -2[f(x)+f(-x)]
g(-x) = g(x) ... the right hand side (RHS) can be replaced with g(x) since g(x) = -2[f(x)+f(-x)]


Conclusion: Since g(-x) = g(x) is definitely true, this proves that g(x) is indeed an even function.
Because g(x) is even, this means it has <a href="http://www.mathwords.com/s/symmetric_y_axis.htm">symmetry with respect to the y axis</a>.