Mathematics - Functions Question with Solution | TestHub

Change x to 10 - x to obtain

$f(20-x)=f(x)$

We have $\mathrm{f}(20-\mathrm{x})=-\mathrm{f}(20+\mathrm{x})$
$\Rightarrow f(x)=-f(20+x)$

Now change x to $20+x$

$\begin{array}{rlr}& f(20+x)=-f(40+x)& -f(x)=-f(40+x)\end{array}$

$\mathrm{f}(\mathrm{x})=\mathrm{f}(40+\mathrm{x})$, so f is periodic

Again $\mathrm{f}(-\mathrm{x})=-\mathrm{f}(20-\mathrm{x})=-\mathrm{f}(\mathrm{x})$

Thus f is odd