In general there are two different types of symmetry that are of interest with functions. Let y=f(x) in this example.
y-axis symmetry: Replace x with -x in the given equation and simplify it. If you get the same equation back, it has y-axis symmetry. Visually, if you were to fold the graph along the y-axis, the two parts look like mirror images of each other.
Example: Try y = 3x[sup:34s7zif3]2[/sup:34s7zif3]+1
symmetry about origin: Replace x with -x and y with -y. If you get the same equation back, it has symmetry about the origin (or with respect to the origin).
Example: Try y=4x[sup:34s7zif3]3[/sup:34s7zif3]-13x
If you are not considering functions, and looking at just a generic graph, you can also look to see if it has x-axis symmetry: To do this, replace y with -y and see if when you simplify the equation algebraically, you get the same equation back. Visually, if you were to fold the graph along the x-axis, they look like mirror images.
Example: Try y[sup:34s7zif3]2[/sup:34s7zif3]=5x+2/3
An equation may have one, two, three, or none of these symmetries. When testing for symmetry, you need to do the work to verify (or disprove) each type of symmetry.
With respect to domain and range, the way I think of it is that the domain is all the values the independent variable, or input, can can take on (typically, but not always, denoted by the x variable), while the range is all the values the dependent variable, or the output, takes on.
Hope this helps.
