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