Question

if
$f(x) = x {}^{2} + 2xsinx + 3$,then prove that f(x) is an even function​

Answer 1

$\Large\textrm{We know,}$

a function is said to be even if it is symmetric to the y-axis.

• $f(x)=f(-x)$

$\Large\textrm{We are given: -}$

• $f(x)=x^{2}+2x\sin x+3$

$\Large\textrm{It follows that: -}$

$f(-x)$

$=(-x)^{2}+2(-x)\sin(-x)+3$

$=x^{2}+2\sin x+3$

$\therefore f(x)=f(-x)$

Hence, an even function.

$\Large\textrm{See more,}$

$\large\textrm{Trigonometric Functions}$

• $\sin(-x)=-\sin x$

• $\cos(-x)=\cos x$

• $\tan(-x)=-\tan x$

$\large\textrm{Properties of Functions}$

• The sum of odd functions is odd.

• The sum of even functions is even.

• The product of odd functions is odd.

• The product of even functions is even.

But, notice that these rules work to sum or product of the same type of functions. The sum or product between two functions may neither be odd nor even.

Answer 2

refer the given attachment