Answer:
f(x)=(g(x))^2 is an even function.
Step-by-step explanation:
We are given that g(x) is an odd function.
we have to determine that among the four options which are an even function.
(a) f(x)=g(x)+2
let g(x)=x ; is an odd function.
Then f(x)=x+2
on computing f(-x)= -x+2
We see that f(-x)≠f(x).
Hence f(x) is not an even function.
Hence, option (a) is incorrect.
(b) f(x)=g(x)+g(x)=2g(x)
as g(x) is an odd function.
so let g(x)=x
f(x)=2x
f(-x)= -2x.
here also we get f(-x)≠f(x).
Hence f(x) is not an even function.
(d) f(x)= -g(x)
Let g(x)=x which is an odd function.
Then f(x)= -x
also f(-x)=x
Here also f(-x)≠f(x)
Hence f(x) is not an even function.
Hence (a),(b) and (d) are incorrect. We are left with option (c).
Also let us consider :
(c) f(x)= (g(x))^2
as square of an odd function is an even function.
Hence f(x) is an even function.
