Question 996033: Given a function, f(x), determine the symmetry of g(x)=-2[f(x)+f(-x)]
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! 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 symmetry with respect to the y axis.
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