Odd and Even functions

Question

if f(x) is a odd function, which of the following must be an even function

a) 2·f(x)

b) x²+f(x)

c) x+f(x)

d)x²·f(x)

e) x·f(x)

Answer 1

Odd and Even functions

if f(x) is a odd function, which of the following must be an even function

a) 2·f(x)

b) x2+f(x)

c) x+f(x)

d)x2·f(x)

e) x·f(x)

Odd functions exist when f(-x) = -[f(x)]

Or basically when you substitute x with -x in the function, if you arrive at the negative of the equation after simplifying. Then it is an odd function. Remember: Odd function are symmetric about the origin.

Even functions exist when f(-x) = f(x)

Or basically when you substitute x with -x in the function, and you arrive at the same equation after simplifying. Then it is an even function. Remember: Even functions are symmetric about the y-axis.

So:

a) 2 - f(x) is an odd function shifted up units two and reflected over x-axis, not even

b) x²+f(x) even function plus and odd function, not even

c) x+f(x) odd + function, still odd

d) x²·f(x) even times odd is still odd because adding 2 two a 3 = 5, still odd

e) x·f(x) odd times odd = even, example: x³ * x⁵ = x⁸ (even)

Answer: e) x·f(x)