Let `G(x)=(1/(a^x-1)+1/2)F(x),` where a is a positive real number not equal to 1 and `f(x)` is an odd function. Which of the following statements is true? `G(x)` is an odd function `G(x)i s` an even function `G(x)` is neither even nor odd function. Whether `G(x)` is an odd or even function depends on the value of a

Options

A. G(x) is an odd function
B. G(x) is an even function
C. G(x) is neither even function nor odd function
D. Whether G(x) is an odd function or an even function, it depends on the value of a

Answer

B

Explanation

`G(x)=((1)/(a^(x)-1)+(1)/(2))F(x)`
`thereforeG(-x)=((1)/(a^(-x)-1)+(1)/(2))F(-x)`
`=-((a^(x))/(1-a^(x))+(1)/(2))F(x)`
`=((a^(x))/(a^(x)-1)-(1)/(2))F(x)`
`=((a^(x)-1+1)/(a^(x)-1)-(1)/(2))F(x)`
`=(1+(1)/(a^(x)-1)-(1)/(2))F(x)`
`=((1)/(a^(x)-1)+(1)/(2))F(x)=G(x)`
`thereforeG(x)` is an even function.