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

Jul 28, 2018

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

#### Explanation:

$\arccos x = \frac{y}{x} \in \left[0 , \pi\right]$

$\Rightarrow$ polar $\theta \in \left[0 , \arctan \pi\right] \mathmr{and}$[ 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,

$x = \cos \left(\frac{y}{x}\right) \in \left[- 1 , 1\right\}$

At the lower terminal, $\in {Q}_{3} , x = - 1$

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

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]}