Even vs. Odd Functions

Question

Determine whether each of the following functions is even, odd, or neither.

A. f(x) = 2x^4 - 6x^2 + 5x - 15

B. f(x) = -4x^7 + 3x^5 + 2x^3 -6x

C. f(x) = 8x^4 - 2x^2 +24

D. f(x)=10x^3 - 15x^2 -16x +12

E. f(x) = 6x^3 + 5x^2 -2x +18

F. f(x) = 4x^5 - 16x^3 + 12x

G. f(x) = 4x^6 + 3x^4 - 2x^2 -12

H. f(x) = x^5 -6x^3 -8x

Answer 1

Test to determine if a function y=f(x) is even, odd or neither:

Replace x with -x and compare the result to f(x).

If f(-x) = f(x), the function is even.

If f(-x) = - f(x), the function is odd.

If f(-x) ≠ f(x) and f(-x) ≠ -f(x), the function is neither even nor odd.

Terms which involve odd powers of x will change signs when x is replaced with (-x).

Terms which involve even powers of x will remain the same when x is replaced with (x).

And since constant terms do not involve x, they will also remain the same when x is replaced with (-x).

For example:

A. f(x) = 2x^4 - 6x^2 + 5x - 15

f(-x) = 2(-x)^4 - 6(-x)^2 + 5(-x) - 15

= 2x^4 -6x^2 - 5x - 15

The resultant equation does not equal f(x) and it does not equal -f(x), so the answer would be neither

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Answer 2

Please see video for a slick shortcut. :-)