Question

If $2f\left(x\right)+3f\left(\frac{1}{x}\right)=x^2-1$, then $f\left(x\right)$ is

• A. a periodic function
• B. an even function
• C. an odd function
• D. even and odd both.

Answer 1

The correct option is B an even function

Given,

$2f\left(x\right)+3f\left(\frac{1}{x}\right)=x^2-1$........(1).

Now putting $\frac{1}{x}$ in stead of $x$ in equation (1) we get,

$3f\left(x\right)+2f\left(\frac{1}{x}\right)=\frac{-x^2+1}{x^2}$.....(2).

Now,

(2)x 3-(1)x 2 we get,

$5f\left(x\right)=-3-\frac{3}{x^2}-2x^2+2$

Or, $f\left(x\right)=-\frac{1}{5}-\frac{2}{5}{x}^2-\frac{3}{5x^2}$.

Now it is clear that,

$f(-x)=f\left(x\right)$.

So the function is an even function.