# Math Help - What does -f(x) mean? Is it the same thing as f(-x)?

1. ## What does -f(x) mean? Is it the same thing as f(-x)?

So, In my precalc class we're learning whether or not a function is odd. And I keep on seeing this function notation -f(x) how do i evaluate it, is it just the opposite of the function? or is it just -1[f(x)]?

And what about if i'm trying to find out if a conic section is symmetrical to the origin, for instance an equation like x^2 + y^2 = 25, how do I do this -f(x) on an equation in which y isn't separated from the rest of the variables?

2. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

Originally Posted by glambeth
So, In my precalc class we're learning whether or not a function is odd. And I keep on seeing this function notation -f(x) how do i evaluate it, is it just the opposite of the function? or is it just -1[f(x)]?

And what about if i'm trying to find out if a conic section is symmetrical to the origin, for instance an equation like x^2 + y^2 = 25, how do I do this -f(x) on an equation in which y isn't separated from the rest of the variables?

I think this should help: Even and odd functions - Wikipedia, the free encyclopedia

3. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

I understand most of the stuff on that aforementioned wikipedia aritcle, however, I don't understand how to evaluate -f(x) of a function. Care to elucidate that to me?

4. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

Originally Posted by glambeth
I understand most of the stuff on that aforementioned wikipedia aritcle, however, I don't understand how to evaluate -f(x) of a function. Care to elucidate that to me?
Yes.

If f(x)=x^2+5x then -f(x)=-(x^2+5x)=-x^2-5x; if g(x)=sin(x) then -g(x)=-sin(x)

5. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

Originally Posted by glambeth
I understand most of the stuff on that aforementioned wikipedia aritcle, however, I don't understand how to evaluate -f(x) of a function. Care to elucidate that to me?
Here is an example.
Suppose that $f(x)=x^4-3x^3-2x^2+x+1.$
Now $-f(x)=-(x^4-3x^3-2x^2+x+1)=-x^4+3x^3+2x^2-x-1.$

Whereas,
$f(-x)=(-x)^4-3(-x)^3-2(-x)^2+(-x)+1=x^4+3x^3-2x^2-x+1.$

6. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

Originally Posted by glambeth
So, In my precalc class we're learning whether or not a function is odd. And I keep on seeing this function notation -f(x) how do i evaluate it, is it just the opposite of the function? or is it just -1[f(x)]?

And what about if i'm trying to find out if a conic section is symmetrical to the origin, for instance an equation like x^2 + y^2 = 25, how do I do this -f(x) on an equation in which y isn't separated from the rest of the variables?
$x^2+y^2=25$

$y^2=25-x^2$

Then, $f(x)=y=\sqrt{25-x^2}$ or $g(x)=y=-\sqrt{25-x^2}.$

$-f(x)= -\sqrt{25-x^2}$

$-g(x)=\sqrt{25-x^2}$

7. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

I like to say that -f(x) is f(x) reflected around the x-axis.

8. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

Originally Posted by pickslides
I like to say that -f(x) is f(x) reflected around the x-axis.
And f(-x) is f(x) reflected about the y-axis.

9. ## Re: What does -f(x) mean? Is it the same thing as f(-x)?

-f(x) is indeed (-1)(f(x)). f(-x) is f applied to -x.

for example: $f(x) = x^2$

$f(-x) = (-x)^2 = (-x)(-x) = x^2 = f(x)$, this function is even.

$-f(x) = (-1)(f(x)) = (-1)(x^2) = -x^2$.

another example: $g(x) = x^3$.

$g(-x) = (-x)^3 = -x^3 = (-1)(x^3) = -g(x)$. this function is odd.