suppose that we have graph of function
$$f(x)=1+x\cos(x)$$
and we should rotate it by $180$ degree,question is what is a function which describe new graph?answer is $$f(x)=x\cos(x)-1$$but i can't understand why it is so?as i know rotation by $180$ is equal to instead of $x$,put $-x$ is not it?in this case we will have $f(x)=1-x\cos(x)$ or we should change y ordinate?i mean when $x=0$,then $y=1$,so we should change sign in $y$ intersection?please help me
$\endgroup$1 Answer
$\begingroup$To rotate $f(x)$ by 180 degrees about the origin, you need to mirror it horizontally ($f(-x)$) and also vertically ($-f(x)$). In your case, $$\begin{eqnarray*} -f(-x) &=& -(1+(-x)\cos(-x)) \\ &=& -(1 - x\cos(-x)) \\ &=& -1 + x\cos(-x) \\ &=& -1 + x\cos(x). \end{eqnarray*} $$
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