Question

how exactly would i do this ? Determine if the following function is even, odd, or neither.

f(x) = -9x^4 + 5x + 3

Answer 1

@welshfella @rock_mit182

Answer 2

I know i have to replace x with -x and see if it stays the same or not but I'm still confused

Answer 3

what did you get after replacing x with -x
that is what is f(-x)=?

Answer 4

I got -9 (-x^4)+5-x+3

Answer 5

\[f(x)=-9x^4+5x+3\]
\[f(-x)=-9(-x)^4+(-x)+3\]

Answer 6

do you know how to simplify (-x)^4?

Answer 7

no... how do you?

Answer 8

do you know what (-1)^4=?

Answer 9

yes, its just 1

Answer 10

so (-x)^4=?

Answer 11

oh just 1

Answer 12

or x

Answer 13

well no you have x inside the 4th power

Answer 14

(-x)^4 is the same as (-1*x)^4
which is the same as (-1)^4*x^4 by law of exponents
and you just said (-1)^4 is 1
so (-x)^4=x^4

Answer 15

ok so now we have:f(−x)=−9(−x)4+(−x)+3

Answer 16

\[(-x)^{even number }=x^{even number} \\ (-x)^{odd number } =- (x^{odd number}) \text{ or we just write this as } -x^{oddnumber}\]

Answer 17

4 is even number so (-x)^4 = x^4

Answer 18

so its even right? since it stayed the same

Answer 19

we haven't got to the other terms yet

Answer 20

oh ok so what do we have so far just so i know whats going on exactly

Answer 21

are you understanding we have only played with the first term in f(-x)?

Answer 22

you do see that (-x)^4 is x^4 ?

Answer 23

yes i do understand

Answer 24

so that is as far as we have gotten so far :p

Answer 25

ok

Answer 26

\[f(x)=-9x^4+5x+3 \\ f(-x)=-9(-x)^{4}+5(-x)+3 \\ f(-x)=-9x^4-5x+3\]

Answer 27

so looking at f(x) and f(-x) do you think they are the same ?

Answer 28

compare the terms
are all the terms the same?

Answer 29

no

Answer 30

the middle term is different correct?

Answer 31

yes

Answer 32

anyways
same=even

if you multiply f(-x) by -1 and get f(x) then is odd

so if you multiply every term in f(-x) by -1 is the result the same as the terms in f(x) ?

Answer 33

like we already said the function f(x) wasn't even because you said f(x) and f(-x) weren't the same

Answer 34

if you multiply every term f(-x) by -1 it would not be the same as the other function

Answer 35

so f is not odd either

Answer 36

there is a short cut if you want to know

Answer 37

before I move on to the short cut
do you understand the answer ?

Answer 38

yes i want to know the short cut

Answer 39

so it is neither? I don't quite understand that/....

Answer 40

well if it isn't even and if it isn't odd
then it is neither odd or even

Answer 41

and why were there 3 functions when we were comparing since i thought there were only two?

Answer 42

f(x) wasn't the same as f(-x) so f(x) was not even
f(x) wasn't the same as -f(-x) so f(x) was not odd

Answer 43

i'm using the definition of even and odd function

Answer 44

i know that but why were there three functions when comparing?

Answer 45

the definition of an even function must satisfy f(x)=f(-x)
the definition of odd function must satisfy f(x)=-f(-x)

Answer 46

since there is only an even and odd

Answer 47

that is why we has to find f(-x) and also -f(-x)

Answer 48

oh so it was the given function, the odd function, and the even funciton>?

Answer 49

our function was neither odd or even

Answer 50

ok, i get that now. So what is the short cut to this?

Answer 51

ok do you know how to find the degree of each term of your polynomial ?

Answer 52

\[f(x)=-9x^4+5x+3\]

Answer 53

sort of, I'm a little bit fuzzy on that though i learned it last year

Answer 54

what is the degree of the first term?
the degree of the second?
the degree of the third?

Answer 55

hint look at the exponent on the variable

Answer 56

1st- 4, 2nd- 2, 3rd-1 ? I'm probably wrong there ...

Answer 57

ok this might help you
our function is equivalent to
\[f(x)=-9x^4+5x^{\color{red}{1}}+3\color{red}{x^{0}}\]

Answer 58

if you see a variable and no power the power is understood to be one

Answer 59

oh ok, that was the other option i was debating in my head. ok, got it.

Answer 60

if there is no variable then the degree is 0

Answer 61

so if you have a polynomial and all the powers are even then the function is even

if you have a polynomial and all the powers are odd then the function is odd

if you have a polynomial and you have a mixture of even and odds in your powers then the function is neither even or odd

Answer 62

the powers were 4,1,0
are all of these numbers even , odd, or do you have a mixture?

Answer 63

oh wow, thats a big help. Thank you so much for sharing the tip! And we have a mixture since 0 is even and 1 is odd

Answer 64

right

Answer 65

so the function f(x)=-9x^4+5x+3 is neither even or odd

Answer 66

what about f(x)=9x^4+3 ?

Answer 67

is would be a mix also since 3 would have degree of 0, and 4 is even

Answer 68

isn't 4 and 0 even?

Answer 69

oh yeah duh wow that was dumb. yes,i knew that. the function would be even.

Answer 70

yep

Answer 71

there is one exception

Answer 72

the function f(x)=0 is actually both even and odd

Answer 73

oh ok

Answer 74

but anyways it will work every other time for any other polynomial function

Answer 75

ok. thanks! That helped a lot. can you help me with another question I have if i tag you in a new post?

Answer 76

the reason f(x)=0 is both even and odd because it does satisfy both definitions

f(-x)=0 so f(x)=f(-x) so f is even
-f(-x)=-0=0 so f(x)=-f(-x) so f is odd

Answer 77

ok

Answer 78

ok!

Answer 79

thanks for the help. Ill tag you now.