# Thread: f(x) vs. f(-x) vs. -f(x)

1. ## f(x) vs. f(-x) vs. -f(x)

Hello!

Could someone elaborate on what the difference is between f(x), f(-x), and -f(x)

Lets use the equation...

$f(x)=5x^3$

so if x=3 then f(x)=(5)(27)=135

and if x=-3 then f(x)=(5)(-27)=-135

what is -f(x) in this scenario? (when x=3)

is it simply $-f(x)=-(5x^3)$ or $-f(x)=-(5\cdot27)$ ?

2. Originally Posted by jonnygill
Hello!

Could someone elaborate on what the difference is between f(x), f(-x), and -f(x)

Lets use the equation...

$f(x)=5x^3$

so if x=3 then f(x)=(5)(27)=135

and if x=-3 then f(x)=(5)(-27)=-135

what is -f(x) in this scenario? (when x=3)

is it simply $-f(x)=-(5x^3)$ or $-f(x)=-(5\cdot27)$ ?
You have it right.

-Dan

3. $-f(x) = -1 \cdot f(x)$

If you're graphing it will be a reflection of f(x) in the x-axis.

In your case you have $-f(x) = -(5x^3)$ and hence [/tex]-f(3) = -(5 \cdot 27)[/tex]

In other words yes

4. For this particular function, $f(x)= 5x^3$ because it involves only an odd power of x (and so is an "odd" function), f(-x) happens to be the same as -f(x). For example, f(-2)= 5(-2)(-2)(-2)= 5(4)(-2)= -40 while -f(2)= -(5)(2)(2)(2)= -5(4)(2)= -40.

However, if the function were $5x^2$, involving x only to an even power (and so an "even" function), we would have f(-x)= f(x). For example, f(-2)= 5(-2)(-2)= 20 which is equal to f(2)= 5(2)(2)= 20.

But most functions are neither "even" nor "odd". For example if f(x)= x+ 2, then f(-x)= -x+ 2 is not equal to either -f(x) nor f(x). For example, f(-2)= -2+ 2= 0 while f(2)= 2+ 2= 4.