What is the domain and range for $y = xcos^-1[x]$ ?

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Answer 1 · 2018-07-28T09:20:08

Range: $[ - pi, 0.56109634 ]$ , nearly.
Domain: ${ - 1, 1 ]$ .

$arccos x = y/x in [ 0, pi ]$

$rArr$ polar $theta in [ 0, arctan pi ] and$ [ pi + arctan pi, 3/2pi ]#

#y' = arccos x - x / sqrt( 1 - x^2 ) = 0, at

$x = X = 0.65$ , nearly, from graph.

y'' < 0, x > 0#. So,

$max y = X arccos X = 0.56$ , nearly

Note that the terminal on the x-axis is [ 0, 1 ].

Inversely,

$x = cos ( y/x ) in [ -1, 1 }$

At the lower terminal, $in Q_3, x = - 1$

and $min y = ( - 1 ) arccos ( - 1 ) = - pi$ .

Graph of $y = x arccos x$
graph{y-x arccos x=0}

Graphs for x making y' = 0:

Graph of y' revealing a root near 0.65:
graph{y-arccos x + x/sqrt(1-x^2)=0[0 1 -0.1 0.1] }
Graph for 8-sd root = 0.65218462, giving

max y = 0.65218462( arccos 0.65218462 ) = 0.56109634:
graph{y-arccos x + x/sqrt(1-x^2)=0[0.6521846 0.6521847 -0.0000001 0.0000001]}

What is the domain and range for y = xcos^-1[x]y = xcos^-1[x]?