If f and g are odd functions, which of the following must also be odd?

Question

1. f(g(x))
2. f(x)+g(x)
3. f(x)g(x)

A. 1 only

B. 2 only

C. 1 and 2 only

D. 2 and 3 only

E. 1, 2, and 3

Answer 1

Hi Woojin,

From the question, we know that for all n:

f(x) = odd function, therefore... f(x) = -f(x) = x²ⁿ⁺¹

g(x) = odd function, therefore... g(x) = -g(x) = x²ⁿ⁺¹

Looking at the options:

I. f[g(x)] then would be equal to f[-g(x)]

--> This is odd because putting an odd function inside an odd function results in an odd function.

II. f(x) + g(x)...

x²ⁿ⁺¹ + x²ⁿ⁺¹ = 2x²ⁿ⁺¹

Therefore x is always an odd function.

III. f(x)g(x)

In this case, we have one odd function multiplied by an odd function, so...

Let f(x)g(x) = x²ⁿ⁺¹ * x²ⁿ⁺¹ = x^2(2n+1)

--> f(x)g(x) is always an even function.

Hope this helps!

-Winn