Question

Explain why domain restrictions for inverse trigonometric functions are necessary

Answer 1

Step-by-step explanation:

Remember that any function pairs each x value with exactly one y value.  No x can have more than one y corresponding to it.

Example: when you take the points on the function y = sin x and reverse their coordinates (switch the x's and y's), you get the blue squiggly line (see graph) that goes up and down the y-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 not a function!  It fails the Vertical Line Test miserably. Each x corresponds to an infinite number of y'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 x will correspond to only one y.

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.