I have $f(x) = \sqrt x$, which means $f^{-1}(x) = x^2$.
I need to determine if the points $(a, f(a))$ where $a \geq 0$ lies on the graph of $f$ or $f^{-1}$. This was easier with points like $(2,4)$ or $(5, \sqrt 5)$ where I could easily plug in the numbers and solve the equation. Does the same hold where I just use the terms $a$ and $f(a)$ to solve the problem?
I suppose this comment evolved to the point of being an answer, so I will make it one:
Taken from Wikipedia: "In mathematics, the graph of a function $f$ is the collection of all ordered pairs $(x,f(x))$." So clearly the general form is correct in your example. If $a$ is in the domain then you can be sure the point is on the graph of $f$.
$(f(a),a)$ would not be on the graph of $f$ except for some specific $a$.