So while practicing some calculus problems for a test, I felt on a relatively small but hard problem that I couldn't even know where to start from :

The problem ask you to prove that the equation of the graph $\Gamma$ is the result of fusionning two graphs $Cg$ and $C_f$ Given that $\Gamma$ 's equation is as follows :

$$ x²y² - 2x²y + x² - 2x + 1 = 0 $$

And that :

$$ f(x) = \frac{x + \sqrt{2x - 1}}{x} $$

$$ g(x) = \frac{x - \sqrt{2x - 1}}{x} $$

And by fusionning two graphs I mean that when two graphs get closer togethr until they form one graph . (I couldn't find the right term in english, given that I study maths generally in arabic ) .

I really don't have any idea on how to approach this kind of problem, I'm not looking for raw answers, I'm looking for explanation and I'd appreciate if you could give an example :) .

  • $\begingroup$ Are you sure there is not a typo in $f$, $g$? Without the factor of 2 outside the square root, $f$ and $g$ would be what you get by solving for $y$ the first equation. In that case the first equation would be a curve in the $(x,y)$-plane which can be written as the superposition of the graphs of $f$ and $g$, with no "fusioning" needed. $\endgroup$ – GFR Oct 21 '16 at 21:12
  • $\begingroup$ @GFR Ouupps, my fault for the 2 edited . $\endgroup$ – Anis Souames Oct 21 '16 at 21:15
  • $\begingroup$ Good! Is my comment clear or should I elaborate? $\endgroup$ – GFR Oct 21 '16 at 21:25
  • $\begingroup$ @GFR So If I understand, I need to solve the 1st equation and I get two solutions, one is f(x) and the other is g(x) ? I'd love if you could elaborate it a bit more in an answer :) . $\endgroup$ – Anis Souames Oct 21 '16 at 21:28

What the graph of a function $f: R \rightarrow R$ really is, is the subset of the plane $R^2$ given by $\{(x,y)\in R^2 :y=f(x))$, or equivalently $\{(x,y)\in R^2 :y-f(x)=0)$. This is not the only way to specify a subset of the plane. For example, I could take a function $F:R^2 \rightarrow R$ and ask you to plot the subset $S$ of $R^2$ given by $S=\{ (x,y)\in R^2 : F(x,y)=0\}$ - this is what you have.

Now not all the equations of the form $F(x,y)=0$ correspond to graphs, but in some cases they do, for example $F(x,y)=y-x=0$. In your case you nearly get a graph, in fact you get two graphs. I will consider a simpler example which has the same features.

Consider the function $F(x,y)= y^2-x^2 $. Which subset of the plane corresponds to the points for which $F(x,y)=0$? Well, solving for $y$ you get $y=x$ and $y=-x$, so the set $\{(x,y)\in R^2 : y^2-x^2=0\}$ is the union of two straight lines through the origin, forming an $x$. Note that the "x" figure formed by this set cannot be the graph of a function $f:R\rightarrow R$ as two values of $y$ correspond to each $x$, but can be written as the union of two graphs.

Finally, in all the examples I solved for $y$ as a function of $x$, but that is not always the best thing to do: sometimes is more convenient to solve for $x$ or to do something else.


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