# What is the difference between f(-x) and -f(x)?

What is the difference between $f(-x)$ and $-f(x)$ in terms of their graphs?

• There's no general answer...imagine g(x) = x^2 or g(x)=x – wxyz Jan 21 '14 at 15:57

The graph of $f(-x)$ is the mirror image of the graph of $f(x)$ with respect to the vertical axis.
The graph of $-f(x)$ is the mirror image of the graph of $f(x)$ with respect to the horizontal axis.

A function is called even if $f(x)=f(-x)$ for all $x$ (For example, $\cos(x)$).
A function is called odd if $-f(x)=f(-x)$ for all $x$ (For example, $\sin(x)$).

The most helpful vocabulary related to your question has to do with the parity of a given function. Functions are described as odd, even, neither. Most functions are neither odd nor even, but it is great to know which ones are even or odd and how to tell the difference.

Even functions - If $f(x)$ is an even function, then for every $x$ and $-x$ in the domain of $f$, $f(x) = f(-x)$. Graphically, this means that the function is symmetric with respect to the $y$-axis. Thus, reflections across the $y$-axis do not affect the function's appearance. Good examples of even functions include: $x^2, x^4, ..., x^{2n}$ (integer $n$); $\cos(x)$, $\cosh(x)$, and $|x|$.

Odd functions - If $f(x)$ is an odd function, then for every $x$ and $-x$ in the domain of $f$, $-f(x) = f(-x)$. Graphically, this means that the function is rotationally symmetric with respect to the origin. Thus, rotations of $180^\circ$ or any multiple of $180^\circ$ do not affect the function's appearance. Good examples of odd functions include: $x,x^3,x^5,...,x^{2n+1}$ (integer $n$); $\sin(x)$, and $\sinh(x)$.

To really get a feel for recognizing these functions, I suggest you graph several (if not all) of them. You will be well-equipped to visually determine the parity of most functions if you just spend a little time graphing the example functions.

• Functions may be neither odd nor even, but having said that, it’s worth pointing out that any function $f$ whose domain allows negation and whose codomain allows addition, negation and division by $2$ can be written as the sum of an odd and even function: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ – the left fraction defines an even function whereas the right fraction defines an odd function. – k.stm Jan 21 '14 at 16:20
• @K.Stm. Interesting observation! I know some properties involving the parity of the result of sums, differences, products, quotients, derivatives, and compositions of various combinations of even and odd functions. But it did not occur to me that you could also decompose a function given the constraints in your comment. Cool! – Xoque55 Jan 21 '14 at 16:26