Which of the following functions is an odd functions ?

Options

A. `f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2))`
B. `f(x)=x((a^(x)+1)/(a^(x)-1))`
C. `f(x)=log_(10)((1-x^(2))/(1+x^(2)))`
D. f(x)=k (constant )

Answer

A

Explanation

If ` f(x)=sqrt(1+x+x^(2))-sqrt(1-x+x^(2))`, then
`f(-x)=sqrt(1-x+x^(2))-sqrt(1+x+x^(2))`
`implies f(-x)=-f(x)`
So, f (x) is an odd functions
Thus, option (a) is correct .
If `f(x)=x((a^(x)+1)/(a^(x)-1))`, then
` f(-x)=x((a^(-x)+1)/(a^(-x)-1))=-x((1+^(x))/(1-a^(x)))=x((a^(x)+1)/(a^(x)-1))-f(x)`
So, f(x) is an even function.
Thus , option (b) is not correct.
If `f(x)=log_(10)((1-x^(2))/(1+x^(2)))`, then `f(-x)=log_(10)((1-x^(2))/(1+x^(2)))=f(x)`
So, f(x) is an even function. Thus, option (c ) is not correct
If f(x)=k for all `x ` then
`f(-x)=f(x) ` for all `ximplies f(x)` is an even functions.