2x^3/6x^2-9, is this even, odd or neither?

Question

campbell_st · Answer

do you know the rules for odd and even functions...

campbell_st · Answer

and it doesn't depend on the value of x.

anonymous · Answer

It just asks is this even, odd or neither

anonymous · Answer

Even and odd functions

campbell_st · Answer

ok... an even function occurs when 

f(-x) = f(x)

and odd function occurs when 

f(-x) = -f(x)

if neither condition is met, the the function is neither odd nor even. 

So substitute x = -x into the function and see what you get..

post the answer

anonymous · Answer

First step. 2x^3/6x^2-9
  -6x^2 +9

campbell_st · Answer

close, it's 

$$f(-x) = \frac{2(-x)^3}{6(-x)^2 - 9}$$ 

can you simplify this..?

anonymous · Answer

(X-3)^2

campbell_st · Answer

not quite 

$$(-x)^3 = -x^3$$

and 

$$(-x)^2 = x^2$$

does that make sense..?

campbell_st · Answer

ok.... so then simplifying you can say 

$$f(-x) = \frac{-2x^3}{6x^2 - 9}$$

which is 

$$f(-x) = -\left( \frac{2x^3}{6x^2 - 9} \right)$$

does that look familiar.... and please scroll up and check to conditions for odd, even and neither functions.

anonymous · Answer

I am confused on how to solve this.

anonymous · Answer

Would it be -2^3/-6x^2+9

campbell_st · Answer

you don't solve it the process is to 

1. replace x with -x

2. simplify where necessary

3. compare it to the original function so see if it's odd or even or neither

anonymous · Answer

It would be neither

campbell_st · Answer

ok... so when I subsituted x = -x, and simplified

I got 

$$f(-x) = - \left( \frac{2x^3}{6x^2 - 9} \right)$$

which means 

$$f(-x) = - f(x)$$

which is the condition for an odd function

anonymous · Answer

F(-x)= 2(-x)^3 turns into -2x^3 then 6(-x)^2 -9 tuns into 6x^2 -9 ; the top number is not the same. For -f(x) = -(2^3/6x^2-9)  = -2x^3/-6x^2+9 so it would be odd rather than neither. Thanks!

campbell_st · Answer

that is incorrect... 

if you $$-1 \times \frac{2x^3}{6x^2 - 9} = \frac{-2x^3}{6x^2 - 9}$$

the denominator is not multiplied by -1, just the numerator... 

its like saying what is 

$$-1 \times \frac{1}{2} = \frac{-1}{2}... or \frac{1}{-2}$$

but remembering the rules for multiplying and dividing negatives

$$\frac{-1}{-2} = \frac{1}{2}$$

which can't occur if you multiply 1/2 by -1

anonymous · Answer

Got it!