Question

show analytically that f(x)=0 is odd and even

Answer 1

what is f(-x) ?

Answer 2

that is why i am here not sure my self but know from reading that this is one of the only functions that is both even and odd, but i have no idea how to show it.

Answer 3

no matter what x you plug in you get the same number back from the function, right?
it's always the same if it is f(x)=0 (or any other constant for that matter)

Answer 4

if
f(x)=0
then
f(1)=f(315)=f(-2)=0
yes?

Answer 5

as i understand it f(1)would be the value to replace x with in an equation and so on for f(315) f(-2) correct?

Answer 6

right, but in the function
f(x)=0
there is no x on the right hand side to replace, so whatever x you plug in changes nothing
understand?

Answer 7

f(x)=0
f(-x)=0(-whatever)=0?

Answer 8

well yeah...
f(x)=0
and
f(-x)=0
hence
f(x)=f(-x)
which means the function is...?

Answer 9

both even and odd

Answer 10

well we have only technically shown one, which one?

Answer 11

even

Answer 12

correct, because we showed that f(x)=f(-x)
so now go ahead and try to prove the oddness yourself

Answer 13

-f(x)=0
=(-1)0(whatever)=0?

Answer 14

not quite, but close
the (whatever) throws me off

Answer 15

what is the main thing we need to show for a function to be odd?

Answer 16

f(x) multiplied by a (-1)

Answer 17

no we need to show that
f(-x)=-f(x)

Answer 18

so find f(-x) and -f(x)
then prove they are equal

Answer 19

we found f(-x)=0
-f(x)=0(-(x)) if we had a x.

Answer 20

how did we know that f(-x)=0 ?

Answer 21

f(x)=0h for the purpose of this disscussion
f(-x)=0(-h)
f(-x)=0
-f(x)=0(-(h))
even since (-(h) is oppisite of our (h)

Answer 22

odd

Answer 23

writing out the stuff (h) in parentheses is making this way confusing...

Answer 24

we know that f(-x)=0 because if we "plug in" ANY number for x into the function we still get 0
hence the negative of any number also gives 0
since f(x)=0, this means that f(-x) also is zero

Answer 25

(yes, I know there is nothing to "plug in" in the equation f(x)=0 but try not to get to worried about that, just use the definitions of even and odd)

Answer 26

since we showed above that f(x)=0 and f(-x)=0
then f(x)=f(-x)
now, we have established that the value of the function is \(independent\) of x...

Answer 27

f(x)=0
f(-x)=-0=0
-f(x)-(0)=0
better

Answer 28

better, but you are an equals sign short of a perfect proof

Answer 29

f(x)=0
f(-x)=0 (no -0 because there is no place to "plug in -x, so don't even use it!)
-f(x)=-(0)=0

Answer 30

well that explains it then have even got to proof yet but would welcome any help!

Answer 31

we need to show two things:
that f(-x)=f(x) and that f(-x)=-f(x)
the first we can show by "plugging in x" as I described above
thing 1:
f(x)=0
f(-x)=0
therefor f(-x)=f(x) (even)

Answer 32

thing 2
f(-x)=0
-f(x)=-(0)=0
but then we observe that this is the same as f(-x), therefor
f(-x)=-f(x) (odd)

Answer 33

ok i see it thank you now just need to get this down mentally, thanks again.

Answer 34

welcome :)