Which functions are odd? I need help

Question

Which functions are odd? Select all that apply. (pick 3)

f(x)= -(1)/(2)x^4+5

f(x)=-8x^3+5x

f(x)=(-4)/(x^3-x+1)

f(x)=(x^5)/(x^4-1)

f(x)= - sqrt2x

f(x)=∛x - x^3

Answer 1

A function,f, is odd if and only if f(-x) = -f(x). We have a list of functions f(x) so we should test them to see if they are odd. I'll test the first one as an example.

f(x)= -(1)/(2)x^4+5

For ease of reading, I will change 1/2 to 0.5 and x^4 to x⁴

f(x)= -0.5x⁴+5

Take the condition f(-x) = -f(x) and plug in your function. f(-x) just means to replace each 'x' in f(x) with '-x'. So the Right Hand Side of the equation is:

f(-x) = -0.5(-x)⁴ + 5

If an exponent is even, that means that the result will always be positive (think of -2*-2, the negatives cancel out and make a positive, and this translates to every even power). This means that (-x)⁴ = x⁴, so:

f(-x) = -0.5x⁴ + 5

And the Left Hand Side of the equation is negative of the original f(x).

-f(x) = - (-0.5x⁴ +5)

Now distribute the negative sign into the parentheses:

-f(x) = 0.5x⁴ - 5

Now we can compare and see if f(-x) = -f(x).

-0.5x⁴ + 5 ≠ 0.5x⁴ - 5

We can see that the equations are not the same, so this is not an odd function.

Try the rest of them and see what you come up with!