# Let `G(x) = 2sin^(-1) (sqrtx)/2` 1) Findthe domain of G(x)2) Find the domain of G'(x) = 1/(sqrt(-(x-4)(x))

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`G(x)= 2sin^(-1) sqrtx/2`

(1) To determine the domain of G(x), we need to consider the properties of a square root. Note that in square root, the radicand should be always greater than or equal to zero.

So, values of x should be `xgt=0` .

Also, we need to consider the doman of the basic function of inverse sine.

The domain of `y=sin^-1x` is `-1lt=xlt=1` .

Base on these two properties, `sqrtx/2` should satisfy the condition `0<=sqrtx/2<=1` .

Then, solve for the values of x.

`sqrtx/2 gt=0` and `sqrtx/2lt=1`

`sqrtxgt=0 ` `sqrtxlt=2`

`x>=0 ` `xlt=4`

**Hence, the domain of G(x) is `0lt=xlt=4` .**

(b) To determine the domain of G'(x), note that in a rational function a zero denominator is not allowed.

Also, since we have a square root in the denominator, the radicand should be greater than or equal to zero.

Considering the two conditions above, we then have:

`-(x-4)x gt 0`

Set each factor greater than zero.

`-(x-4)gt0 ` and `xgt0`

`x-4lt0`

`xlt4`

**Thus, the domain of G'(x) is `0ltxlt4` .**