Question

How do you find the domain and range of `f(x) = ( tan(2x) ) / ( (sin^-1)(x)- pi/3)`?

Answer 1

Domain: `x in [ - pi/2, 1/2sqrt 3 ) U ( 1/2sqrt3, - pi/4)`
`U (- pi/4, pi/4 ) U ( pi/4, pi/2]`
Range: ( - oo, oo )#

Explanation:

The presence of `sin^(-1)x` restricts the domain to be

( - pi/2, pi/2 ). sans asymptotic x, within.

`y = (tan 2x)/(sin^(-1)x - pi/3 ), x ne` asymptotic 1/2sqrt3, +-pi/4#.

Range: `y in ( - oo, oo )`

See graph.
graph{(y( arcsin (x ) - pi/3 ) - tan (2x ))(x-1/2sqrt3-0.0001y) = 0[-6 6 -3 3]}