Question

What is the domain of `g(x) = sin^-1(2x + 3)`?

Answer 1

`x in [ -2, - 1 ]` and `y in [ -pi/2, pi/2 ]`, for this truncated graph.

Explanation:

`g ( x ) = sin^ ( - 1 ) ( 2 x + 3 )`.

This is one-piece inverse with

`g in [ - pi/2, pi/2 ]` of `x = 1/2( sin g - 3 )`.

Correspondingyt, the domain is given by

`x in [ 1/2 ( sin ( - pi/2 ) - 3), 1/2 ( sin ( pi/2 ) - 3 ) = [ - 2, - 1 ]`.

See illustrative graph, within the enclosure

`x = -2, y = pi/2, x = - 1 and y = -pi/2`o.

graph{(y - arcsin ( 2 x + 3 ))(y^2-(pi/2)^2) = 0[-3 0 -2 2]}

For information, the wholesome graph for

`g = (sin)^( - 1 )( 2x + 3 )`, using the inverse `x = 1/2 sin (( g ) - 3 )`

is shown below.

graph{x-1/2( sin (y) - 3 ) = 0 [-3 0 -10 10]}

Here, g-range is without limit.

I use `(sin )^( - 1 )` for the wholesome inverse. This enables me to

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