Determine if the following functions are odd, even, or neither?

Question

f (x) = (x)^2 -x , and f (x) = (2x)^2 + (x)^4 + 1

need help plz...if its not to much to ask... a little explanation would help me alot plz and ty

Answer 1

even means symmetric to the y-axis. f ( x ) = f ( -x )

odd means symmetric to origin: f ( x ) = - f ( -x )

those are the rules, so...

1) f(x) = x^2 - x

even test:

f(-x) = (-x)^2 - (-x)  = x^2 + x not equal to f(x) -----> test fail, NOT even

odd test:

since I already did f(-x) just negate it to find -f(-x)

-f(-x) = -x^2 - x not equal to f(x) -------> NOT odd, so neither.

2) f(x) = 2x^2 + x^4 + 1

even test:

f(-x) = 2(-x)^2 + (-x)^4 + 1 = 2x^2 + x^4 + 1 = f(x) -----> YES! it's even!

ps: I assume you meant 2x^2, not (2x)^2. of course it's possible it's written like that, but not likely. either way it's still even

Answer 2

The test is: substitute x by -x in the function and

a) if it has no change it is even

b) if it only changed the signal it is odd

d) neither in other cases.

1) f(-x) = (-x)^2 -(-x) = x^2 + x it changed only part of the signal... neither.

2) f(-x) = (-2x)^2 + (-x)^4 + 1 = (2x)^2 + x^4 + 1 ... no change... even.

another example: f(x) = x^3 -3x

f(-x) = (-x)^3 -3(-x) = -x^3 + 3x = -(x^3 -3x)... only changed the signal... it is odd.