Question

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

Answer 1

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

Explanation:

`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]}