Question 439239

here is y = cos^-1x = arccosx and its graph:

open this image:

http://img23.imageshack.us/f/arccos3.gif/




Since {y = cos^-1(x) is the {{{inverse}}} of the function {{{y = cos x}}}, the function y = cos^-1(x){{{if}}} and {{{only}}}{{{ if}}}{{{ cos y = x}}}. 
But, since {{{y = cos x}}} is {{{not}}} {{{one-to-one}}}, its {{{domain}}} must be restricted in order that y = cos^-1(x) is a function.

To get the graph of y = cos^-1x, start with a graph of {{{y = cos x}}}.



*[invoke plot_any_graph "cos(x)", -4, 4, -4, 4, 500, 500]

{{{Restrict}}} the {{{domain}}} of the function to a one-to-one region - typically [{{{0}}}, {{{pi}}}] is used for cos^ -1x. This leaves the range of the restricted function unchanged as [{{{-1}}}, {{{1}}}].



Reflect the graph across the line {{{y = x}}} to get the graph of y = cos^-1 x y = arccos x, the black curve at right on the graph above.



Notice that y = cos^-1x has domain [{{{-1}}},{{{ 1}}}] and range [{{{0}}}, {{{pi}}}]. It is strictly decreasing on its entire domain.