Question

Let f be an odd function with domain that contains 0. Show that f(0)=0
Let g(x)=f(x)-f(-x)
a. Show that g is an odd function
b. If f is even show that g(x)=0 for all numbers x.

Answer 1

well use g(x)=f(x)-f(-x)
what is g(0)?

Answer 2

not sure

Answer 3

well if you replace x with 0
you have g(0)=f(0)-f(-0)
...
we can find g(0) easily
but then we need to somehow find that f(0)=0

Answer 4

major hint:
recall if f is odd
f(-x)=-f(x)

Answer 5

another hint: you want to know what happens when x is 0

Answer 6

is there 3 questions here?
or are you suppose to use the 2 questions beneath to show f(0)=0?

Answer 7

you don't need those ...

Answer 8

just two questions

Answer 9

but anyways again replace x with 0
f(-x)=-f(x)

Answer 10

I see 3 questions

Answer 11

show f(0)=0
show g is odd
f is even then show g=0 for all x

Answer 12

ohh ok. Can you help me answer them

Answer 13

trying to but you haven't used what I said to show f(0)=0

Answer 14

you have f is odd
so f(-x)=-f(x)
and you want to know what happens at x=0 so plug in 0 for x

Answer 15

anyways I guess I will move on to the next one...
hint for "show g is odd"
g(x)=f(x)-f(-x)
to show g is odd you need to show g(-x)=-g(x)
so plug in -x and show that

Answer 16

hint for the last one:
if f is even then g=0 for all x

well recall g(x)=f(x)-f(-x)

and since f is even then f(-x)=f(x)