Question

determine whether the functions even, odd or neither. f(x)= x^2sinX. can any one explain how to check if this is even or odd pls.

Answer 1

Good question.
x^2 is even, it is symmetric around x = 0 and so (-1)^2 = 1^2, for example
sin x is odd, it is not symmetric, but sin x = - sin (-x)
OK so far?

Answer 2

put -x

(-x)^2sin(-X) = -x^2sinX

sign changes so odd
otherwise even

Answer 3

Yes. f(-x) = - f(x), so x^2 sin x is odd.

Answer 4

got it thank u!

Answer 5

You're welcome

Answer 6

is this right? f(x)= x^3+cosX
f(-x)=(-x)^3+cos(-x)-------->(-X^3)+(-cosX)----->so f(-x)= -X^3-cosX thus this is odd because the sing changes??

Answer 7

absolutely correct :)

Answer 8

why could the answer be neither?

Answer 9

oops, my badd

f(-x)=(-x)^3+cos(-x)-------->(-X^3)+(+cosX)----->so f(-x)= -X^3+cosX

neither

Answer 10

gtg

Answer 11

Think about what happens to
f(x)= x^3+cosX
for negative x

Answer 12

take x = pi/4
f(pi/4) = (pi/4)^3 + 1/sqrt(2)
f(-pi/4) = -(pi/4)^3 + 1/sqrt(2)
so f(x) is not equal to f(-x) but neither is it equal to -f(-x)

Answer 13

I dont understand how we get a +x when we cube a -x or how we find the x from -x...

Answer 14

(-x)^3 = (-1)^3 x^3 = -1 x^3 = -x^3

Answer 15

make that last -(x^3)

Answer 16

hum I mean is see -(x^3) the same as -x^3...Im sorry, I don't get it. =[

Answer 17

An even function is a function where f(-x) = f(x).
So x^2 is an even function. OK?

Answer 18

And an odd function is a function where f(-x) = -f(x).
But some functions are neither even nor odd. OK?

Answer 19

Try a product like
f(x) = x^2 cos x
(-x)^2 = x^2
cos x = cos -x
So
f(-x) = (-x)^2 cos -x = x^2 cos x = f(x) so it's even. OK?

Answer 20

Lost you I guess.

Answer 21

I understand how the powers work but like on cos x= cos -x we are changing it to f(-x) which I see but I don't see how it ends up being positive--cosx

Answer 22

Umm

Answer 23

If you look at the figure for the cosine function, do you see how it is symmetrical across the y axis?

Answer 24

yes

Answer 25

That symmetry is exactly the same as saying that cos(x) = cos(-x)

Answer 26

Or look at this figure, I just grabbed it quickly so it's not properly labeled but still.

Answer 27

A negative angle h is a rotation below the x-axis. But the cosine is still positive.

Answer 28

ok I see, have a better idea now. thank you!

Answer 29

yw