Question

Determine algebraically whether the function is even, odd, or neither even nor odd.

f(x) = -4x3 + 4x



Neither

Even

Odd

Answer 1

I'm really confused. Can you help explain the concept please.

Answer 2

a function is odd iff f(-x)=-f(x) and is even iff f(-x)=f(x)

so start by finding f(-x)=-4(-x)^3+4(-x)=4x^3-4x=-[-4x^3+4x]=-f(x)

therefore f is an odd function

Answer 3

Okay thanks but in lamest terms, does it have anything to do with the exponents?

Answer 4

hmmmm, odd functions have odd-leading exponents, even ones have, well even-leading exponents
yours has a leading term of 3, so is 3rd degree polynomial

Answer 5

right off, yours is an odd, yes, but you're being asked to do it as kelvinltr showed, by checking for symmetry

Answer 6

usually if all exponents are odd, then the function is odd and if all exponents are even, then the function is even. I'm not sure whether that is a theorem though. Plus guess that works only for polynomials

In the meantime, if you can, try to find a function which is both odd and even at the same time :)

Answer 7

though my understanding of symmetry is that to check for origin symmetry, you set y and x to negative values,, and should get an exact f(x)

Answer 8

The question said "algebraically"....

Answer 9

mind you that all odd functions are symmetric to the origin

Answer 10

So if the equation had a graph, we would be able to draw a line of symmertry?

Answer 11

well, the equation does have a graph, but you're not asked to do it graphically
but yes yes, the line of simmetry is the y=x line

Answer 12

Ahhh okay thank you. So this proves that the function is odd; however, can I determine the ood/even from the exponents every time?

Answer 13

sure

Answer 14

What about the neither?

Answer 15

THAT's what you're being asked this time :)

Answer 16

I'm lost.

Answer 17

Okay. Thanks for all your help guys. I appreciate it.

Answer 18

so, lemme test for symmetry to the origin, as I know is an odd and thus it will have symmetry to the origin
to test for symmetry to the origin, you set "y" and "x" to negative both
so
$$\bf
f(x) = -4x^3 + 4x \implies y=-4x^3 + 4x \\
\text{ x and y to -x and -y}\\
(-y) = -4(-x)^3 + 4(-x)\\
-y = -4(-x)(-x)(-x)+4(-x) \implies -y = 4x^3-4x\\
y = -4x^3+4
$$

Answer 19

so notice, once we set the "x" and "y" to -x and -y
low and behold!
our resultant equation RESEMBLES the original equation
so the function has symmetry to the origin, thus is odd

Answer 20

to test for origin symmetry, you set x and y to negative
if the resultant equation is the same as the original, then you have such symmetry

to test for y-axis symmetry, you set x to negative
if the resultant equation is the same as the original, then you have such symmetry

to test for x-axis symmetry, you set y to negative
if the resultant equation is the same as the original, then you have such symmetry

Answer 21

When your total or anwser is the same as your equation it proves that you have an odd function right?

Answer 22

when the RESULTANT equation, is the same as the ORIGINAL equation
after you've set "x" or "y" or both to negative
THEN you get THAT symmetry
I tested for symmetry to the origin, setting "x" and "y" to negative
and thus it has

Answer 23

Okay thanks. Can you help me with this one: Give an example of a function that is neither even nor odd and explain algebraically why it is neither even nor odd.

Answer 24

I guess a rational will fit that criterion

Answer 25

Can we go a bit futher with the explanation please?

Answer 26

ahemm, I guess I meant an irrational function

Answer 27

but I won't go there

Answer 28

you see a function like \(\bf x^3\)

can be shifted say... upwards by say making it \( \bf x^3+3\)


and thus it will no longer have the origin-symmetry anymore and thus it'd be neither
odd nor even