If y=f(x), then solve for y. If you're asking about things that don't have nice explicit forms, eg a 3rd degree polynomial in y, then yes, the TI-89 can solve those for you.
Sometimes, but not always, you can write implicitely defined functions in terms of explicit functions of a parameter. For example, you can graph x^{2}+ y^{2}= 1, a circle, in parametric graphing mode with x= cos(t), y= sin(t).
Now the TI-89 is programmable so you might search the TI site for packages you can upload to expand its capacities or write your own. At worst you can graph the inequality f(x,y) < 0 by iterating over the coordinates on the visible screen and plotting the ones satisfying the inequality. The boundary will then be your curve.
Yes, if you want to graph an implicit equation without having to first solve for a variable, you could easily write a program to solve it and input it into Y1.
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