Step-by-step explanation:
Remember that any function pairs each <em>x</em> value with <u>exactly one</u> <em>y</em> value. No <em>x</em> can have more than one <em>y</em> corresponding to it.
Example: when you take the points on the function y = sin x and reverse their coordinates (switch the <em>x</em>'s and <em>y</em>'s), you get the blue squiggly line (see graph) that goes up and down the <em>y</em>-axis. For instance, a point on the graph of y = sin x is (pi/2, 1) and this becomes the point (1, pi/2) on the blue wave.
The blue wave is <u>not</u> a function! It fails the Vertical Line Test miserably. Each <em>x</em> corresponds to an infinite number of <em>y</em>'s.
Going back to y = sin x, you restrict its domain (all real numbers) to a set of numbers that will prevent the inverse from "doubling back" on itself--each <em>x</em> will correspond to only one <em>y</em>.
There are standard ways of doing this for each trigonometric function. For the sine function, you restrict its domain to the interval [-pi/2, pi/2}. When the coordinates of the restricted graph are switched, you get the red graph, which is a function, the inverse sine function.