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

Question

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

Answer 1

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, $\sin(x)$).

Answer 2

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.