find natural domain and range for each function:
a) G(x)=(x^2-2*x+5)^(1/2)
b)h(x)=1/(1-sin(x))
c)H(x)=(sin(x^(1/2)))^(-2)
What you posted is not what I consider detailed working.
a) Domain: Solve $\displaystyle x^2-2x+5 \geq 0$.
b) Domain: All real numbers except those satisfying 1-sin(x) = 0.
c) Domain: All positive real numbers except those satisfying $\displaystyle \sin(\sqrt{x}) = 0$.
The details are left for you. I suggest you get ranges by drawing graphs. For a) note that $\displaystyle y=(x^2-2x+5)^{1/2} \Rightarrow y^2 = x^2 - 2x + 5$ and you should recognise a branch of a hyperbola.