How can you apply the rule f(-x)=-f(x) when solving equations?

[trip7]

How do you use f(-x)=-f(x)?

I understand how to use f(x)=f(-x)
Ex:
y=x^3
=(-x)^3
=(-x)(-x)(-x)
=-x^3

but I am confused with the previous.

Any help and examples would be appreciated!

trip7

 

[matt grime]

what on Earth do you mean by "use"?

 

[arildno]

What are you talking about??
What do you mean by "using"?
The example used is wrong for all other choices than x=0.

f(x)=f(-x) is the property of an even function (defined on some interval -a<=x<=a, a>0).

f(-x)=-f(x) is the property of an odd function (defined on some interval -a<=x<=a, a>0).

Usually, exercises involving even/odd require us to determine whether a given function is either even or odd (or neither or both).

Let us take the case f(x)=x^3, and we assume it is defined on the whole number line.
We will investigate whether f is even or odd:

By the definition of f(x), we have:
f(-x)=(-x)^3 (just plugging in -x at x's place in the expression for the function)
We now manipulate the right-hand side:
(-x)^3=(-1)^3*x^3=-x^3
But, x^3=f(x), and we therefore have:
f(-x)=-f(x)
That is, f(x)=x^3 is is seen to be an odd function

 

[Gonzolo]

f(-x) = -f(x) = (-1)*f(x)

 

[trip7]

Pardon my lack of proper terminology. I am trying to find even, odd, neither, or both. What I meant by 'using' is arildno's example "just plugging in -x at x's place in the expression for the function." I am trying to learn curve tracing and checking for symmetry I replaced x with -x aka f(x)=f(-x). I forgot how to find the origin and for some reason f(-x)=-f(x) comes to mind. Is there a simple technique like replace x with -x? Hope this isn't more confusing than previous post.

trip7

 

[trip7]

According to Gonzolo's equation, it appears that you just multiply all terms by -1. Is that correct? I really appreciate everyone help!

trip7

 

[arildno]

Pardon my lack of proper terminology. I am trying to find even, odd, neither, or both. What I meant by 'using' is arildno's example "just plugging in -x at x's place in the expression for the function." I am trying to learn curve tracing and checking for symmetry I replaced x with -x aka f(x)=f(-x). I forgot how to find the origin and for some reason f(-x)=-f(x) comes to mind. Is there a simple technique like replace x with -x? Hope this isn't more confusing than previous post.

trip7

"Im trying to learn curve tracing and checking for symmetry I replaced x with -x aka f(x)=f(-x)."

This is COMPLETELY, TOTALLY WRONG!
Read what I wrote carefully:
First, I EVALUATE f(-x)
Then, I COMPARE the gained expression with what I know of f(x)
Then I can draw a CONCLUSION if f is either odd, even or neither or both.

Another example:
Let g(x)=x^2
Then, we EVALUATE: g(-x)=(-x)^2=(-1)^2*x^2=x^2
But, COMPARING with g(x), I see that I can CONCLUDE:
g(-x)=x^2=g(x), that is, g(-x)=g(x)
That is, g is an even function.

Yet another example:
Let h(x)=x^2-x
Then:
h(-x)=(-x)^2-(-x)=x^2+x

But, the expression x^2+x is not equal to h(x), nor is it equal to -h(x)!
That is:
h(x) is neither even or odd!

 

[trip7]

Thanks for the extra examples arildno, I get it. I put way more into it than I should have!