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If I have 2 equations $y=f(x)$ and $w=g(y)$ then I can nest them to get $w=g(f(x))$. Can a similar thing be done with implicit functions?

Suppose I have a 2 equations $F(x, y)=0$ and $G(y, w)=0$. Can I combine them to get an equation relating $x$ and $w$?

Amaar
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1 Answers1

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It's possible to do "heightmap substitution" $y=f(x)$ to $F(x,y)=0$ to get $F(x,f(x))=0$. This allows calculating $f(x)$ which is $f : (x)\mapsto(y)$

Then $w=g(y)$ can be done for $G(y,w)$ to get $G(y,g(y))=0$ and calculate $g(y)$, which is $g : (y)\mapsto(w)$. Then you just have functions $f:R\rightarrow R$ and $g:R \rightarrow R$ which can be normally composed to get $g(f(x)) : (x) \mapsto (w)$.

tp1
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  • Wouldn't this required "unravelling" F(x, f(x)) = 0 in order to get an expression for f(x)? – Amaar Oct 18 '19 at 22:22
  • yes, you need to know F expression. also f(x) might not be able to give all the values available in F, i.e. you might need to use $f_1(x)$ and $f_2(x)$ if you have circle in F. – tp1 Oct 18 '19 at 22:28