Even and Odd Functions

[AJ22]

Question on if the following function is odd or even:

4x/7x^2 - 1
*the 4x is on the numerator and the 7x^2 - 1 is in the denominator.
Using the approach of f(x) = f(-x) I can identify this function is not even.
f(-x) = -4x/7x^2-1 not equal to 4x/7x^2-1
However, to see if its an odd function I would take f(-x) and that factor out the negative from that answer. I am not too sure if every sign would change if I am to factor out the negative such as:
-f(x) = -(4x/-7x^2 + 1)?

Idk if that is how I would change all the terms but by doing so it is not equal to f(-x) so does that mean the function is neither odd or even?

 

[LCKurtz]

It might help if you would use parentheses properly for the denominator as 4x/(7x^2-1). You should determine it is an odd function. Just replace x by -x and see what happens.

 

[AJ22]

That would make it -4x/(7x^2-1). To see if its odd I would have to factor out -1 which is what I'm confused on.

 

[pka]

4x/7x^2 - 1
*the 4x is on the numerator and the 7x^2 - 1 is in the denominator.
Using the approach of f(x) = f(-x) I can identify this function is not even.
f(-x) = -4x/7x^2-1 not equal to 4x/7x^2-1

We are given that [imath]f(x)=\dfrac{4x}{7x^2-1}[/imath]
Now [imath]f(-x)=\dfrac{4(-x)}{7(-x)^2-1}=\dfrac{-4x}{7x^2-1}=-\dfrac{4x}{7x^2-1}=-f(x)[/imath]


[imath][/imath]

 

[AJ22]

Why doesn't the signs of 7 and -1 change? When you factor out the negative doesn't it change all the signs?

 

[lev888]

Why doesn't the signs of 7 and -1 change? When you factor out the negative doesn't it change all the signs?

We don't factor out -1, it disappears during squaring: (-x)^2 = x^2.
Edit: Which -1 are you referring to? The one in the numerator? It's not supposed to affect the denominator. See post #7.

 

[pka]

Why doesn't the signs of 7 and -1 change? When you factor out the negative doesn't it change all the signs?

Do you understand basic pre-algebra?
Do you understand this: [imath]\dfrac{-4}{7}=-\dfrac{4}{7}=\dfrac{4}{-7}~?[/imath]
If not, there is no wonder that you are confused.

 

[AJ22]

So when your factoring out the negative it would only affect the numerator in this case. So the terms in the denominator would remain untouched:
-(4x/(7x^2 -1))
So if the question had a negative in the denominator like:
4x/(-7x^2-1)
this would be the same as saying -4x/(7x^2-1)?

 

[pka]

So when your factoring out the negative it would only affect the numerator in this case. So the terms in the denominator would remain untouched:
-(4x/(7x^2 -1))
So if the question had a negative in the denominator like:
4x/(-7x^2-1)
this would be the same as saying -4x/(7x^2-1)?

Well there is really no factoring to these. It is a matter of knowing how signs work.
[imath]\dfrac{-4x+3}{7}=\dfrac{-(4x-3)}{7}=-\dfrac{4x-3}{7}[/imath]

 

[lev888]

So when your factoring out the negative it would only affect the numerator in this case. So the terms in the denominator would remain untouched:
-(4x/(7x^2 -1))
So if the question had a negative in the denominator like:
4x/(-7x^2-1)
this would be the same as saying -4x/(7x^2-1)?

Are you saying -7x^2-1 = -(7x^2-1)?
What do you get if you start with -(7x^2-1) and distribute -1?

 

[AJ22]

Well there is really no factoring to these. It is a matter of knowing how signs work.
[imath]\dfrac{-4x+3}{7}=\dfrac{-(4x-3)}{7}=-\dfrac{4x-3}{7}[/imath]

What if I had the 7 in the numerator and the -4x + 3 in the denominator. Would the procedure be the same and would I still end up with the same final answer? By factoring out -1.
7/(-4x+3)
-(7/(-4x+3))

 

[Steven G]

When you factor out a negative sign from you do not get -7 as 7 does not equal -7

If you factor out a negative sign from 7, you'll get -(-7).

Remember that factoring does not change the value of the expression! 7 does equal -(-7) !!!

 

[pka]

What if I had the 7 in the numerator and the -4x + 3 in the denominator. Would the procedure be the same and would I still end up with the same final answer?
By factoring out -1. 7/(-4x+3)

First rewrite [imath]-4x+3[/imath] as [imath]-(4x-3)[/imath] which is more about knowing about how signs work than it is about factoring.
Then rewrite [imath]\dfrac{7}{-4x+3}[/imath] as [imath]\dfrac{7}{-(4x-3)}=\dfrac{-7}{4x-3}~.[/imath]