Question

Which of the following is an even function need help quick

Answer 1

not sure but it is 3..

Thanks!

Answer 2

The correct option is D.

Explanation:

If function f(x) is called an even function if f(-x)=f(x).

The first function is

`F(x)=2\sin((1)/(2)x)`

Put x=-x,

`F(-x)=2\sin((1)/(2)(-x))=-2\sin((1)/(2)x)=-F(x)\\eq F(x)`
`[\because \sin(-x)=-\sin x]`

So, this function is not an even function.

The second function is

`F(x)=2\cos((1)/(2)x+(\pi)/(2))`

Put x=-x,

`F(-x)=2\cos((1)/(2)(-x)+(\pi)/(2))\\eq F(x)`

So, this function is not an even function.

The third function is

`F(x)=2\sin((1)/(2)x+\pi)`

Put x=-x,

`F(-x)=2\sin((1)/(2)(-x)+\pi)\\eq F(x)`

So, this function is not an even function.

The fourth function is

`F(x)=2\cos((1)/(2)x)`

Put x=-x,

`F(-x)=2\cos((1)/(2)(-x))=2\sin((1)/(2)x)=F(x)`
`[\because \cos(-x)=\cos x]`

So, this function is an even function.

Hence the correct option is D.