Which of the following options is an even function Fx 2sin

Which of the following options is an even function? F(x) = 2sin (1/2 x) F(x) = 2 cos(1/2 x + pi/2) F(x) = 2 cos(1/2 x) F(x)= 2sin(1/2 x + pi)

Solution

function is even if

f(-x)=f(x)

1)f(x)=2sin((1/2)x)

f(-x)=2sin((1/2)(-x))

f(-x)=-2sin((1/2)x)

f(-x)=-f(x) not even function

2)

f(x)=2cos((1/2)x +(pi/2))=-2sin((1/2)x)

f(-x)=-2sin((1/2)(-x))

f(-x)=2sin((1/2)x)

f(-x)=-f(x) not even function

3)

f(x)=2cos((1/2)x)

f(-x)=2cos((1/2)(-x))

f(-x)=2cos((1/2)x)

f(-x)=f(x) even function

4)

f(x)=2sin((1/2)x +pi)=-2sin((1/2)x)

f(-x)= -2sin((1/2)(-x))

f(-x)= -(-2sin((1/2)x))

f(-x)=2sin((1/2)x)

f(-x)=-f(x)  not even function