The range of the radical function $y=(x+1)^{1/2}$ is $[0,∞)$. I know how to find it; however, when we solve for $x$ we find the range of $x=y^2-1$ is $[-1,∞)$. So, why do we have two different ranges?
Also, the function y=$x^{1/2}+1$ has a different range from $x=y^2-2y+1$. How is that range differs when we solve for $x$ in both cases?
The range of the inverse function is the domain of the function.
Similarly, the range of the logarithmic function is $\mathbb{R}$, but its domain is $(0,\infty)$, which is the range of the exponential function.
Domain and range of a function are not related (except for the fact that they are specific of the given function, of course).