Difficulty understanding the inverse of a function Let $f(x) = x^2$. Then, the inverse of the function is given by $g(y) = \sqrt y$. When I try graphing $f(x)$ and $g(y)$, the output produces the parabola for $f(x)$ and the other half of the parabola where $x > 0$ for $g(y)$. Why isn't the graph outputting the desired result of $\sqrt x$?

The function $a(x)=x^2\;$is not one-to-one, so it doesn't have an inverse.

The function $b(x) = x^2,\;x\ge 0\;$ _is_ one-to-one, and its inverse is $c(x)=\sqrt{x}$.

Note that the graph of $b\;$is just the right half of the parabola $y=x^2$.

As regards the Desmos graph of $c(y)=\sqrt{y}$, since you used $y$ as the name of the independent variable, Desmos chose to use the $y$-axis as the input axis, and the $x$-axis as the output axis. Thus, the positive input direction is "up", not "right", and the positive output direction is "right", not "up". That only affects the visualization, not how $c$ maps inputs to outputs. To see the graph of $c$ in the usual way, just graph $c(x)=\sqrt{x}$.

If you have more questions, please ask, and I'll try to explain further.