Question

If f(x) is an odd function, which function must also be odd? Explain.
(1) f(x – 1) + 5
(2) 2f(x) + 3
(3) 1/2f(x)
(4) f(x – 4)

Answer 1

review what it means to be odd:
f(-x) = -f(x)
now, which choice fits?

Answer 2

is it 4?

Answer 3

nope.
for #4, that is just the graph of f(x) shifted to the right by 4. So it is no longer symmetric about the origin.
It's #3.
If g(x) = 1/2 f(x) then since f(-x) = -f(x),
g(-x) = 1/2 f(-x) = 1/2 (-f(x)) = - (1/2 f(x)) = -g(x)
so g(x) is also odd