Question

A function $f:R\to R$ satisfying the following conditions
(i) $f(-x)=f\left(x\right)$
(ii) $f(x+2)=f\left(x\right)$
(iii) $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$ and $g\left(1\right)=\alpha$,

Then, the function $g\left(x\right)$ is?

• A. An even function.
• B. An odd function.
• C. Neither even nor odd.
• D. None of the above.

Answer 1

The correct option is B An odd function.

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

$f(x+2)=f\left(x\right)$

From above these two conditions we can say that $f\left(x\right)$ is a constant function.

Now,

$g\left(x\right)={\int }_0^{x}f\left(t\right)dt$

$g\left(x\right)={\int }_0^{x}kdt=kx$
$g\left(x\right)=kx$

$g\left(1\right)=k=\alpha$
And $g\left(x\right)=\alpha x$
$g(-x)=-\alpha x$
$g\left(x\right)=-g(-x)$

$g\left(x\right)$ is an odd function.

Hence, option B.