Question

If g(x) is an odd function, which function must be an even function?

f(x) = g(x) + 2
f(x) = g(x) + g(x)
f(x) = g(x)2
f(x) = –g(x)

Answer 1

f(x)=(g(x))^2

Explanation:

Answer 2

f(x)=(g(x))^2 is an even function.

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.