Assume I have the function.$$f(x)=x^{2}+b$$ where $b$ can be determined by the expression: $$(x-b)+3b^{2}(x^{2}-b^{3})=0$$ Assuming there is no way to isolate the variable $b$ and plug it back into $f(x)$, what would I be able to type into my calculator that graphs $f(x)$ on the x and y 2D argand plane?
For example, when $x=1$, I can find the value of $b$ as $$(1-b)+3b^{2}(1-b^{3})=0$$ $$\Rightarrow b=1$$ Hence:$$f(1)=1^{2}+1=2$$ Since it can be done for individual values of $x$ I assume it can be done for the entire $f(x)$. Is there any way I can make $f(x)$ depend on the other expression so that I can graph it?
EDIT: I realized that I could graph $$(x-y)+3y^{2}(x^{2}-y^{3})=0$$ with each coordinate being the a value of $x$ that satisfies $b$ (as $y=b$ was substituted above).Is there a method I could use to substitute each coordinate back into $f(x)$?