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Odd Funcion or Not?

Source: 1984 National High School Mathematics League, Exam One, Problem 5

February 22, 2020
function

Problem Statement

If a>0,a≠1a>0,a\neq1, F(x)F(x) is an odd function. G(x)=F(x)⋅(1ax−1+12)G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2}), then G(x)G(x) is (A)\text{(A)} odd function (B)\text{(B)} even function (C)\text{(C)} not odd or even function (D)\text{(D)} not sure
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