Describe the preimage of any point $(b,c) \in \mathbb{R}^2$ for $F(x, y) = (x+y,xy)$

Question

I plotted the line and the hyperbola implied by the first and second coordinates respectively. I realize that the preimage can be

• Empty for $c > 0 \land b < c$

• Size one for $c = 0 \lor (b = c \land c > 0)$

• Size two for $c < 0 \lor b > c$.

But I need help actually writing down the expressions for each scenario. I would also appreciate knowing what course/area I should look into to get more practice with this type of problems.

Answer 1

You are looking for solutions to the equation $f(x,y)=(b,c)$, i.e.

$$\left\{\begin{matrix} x+y=b \\ xy=c \end{matrix}\right.$$

Consider two cases

(1) $c=0$. Then $xy=0$ meaning either $x=0$ or $y=0$. The first equation then implies that either $y=b$ or $x=b$. Meaning for $c=0$ there are two solutions: $(0, b)$ and $(b,0)$.

(2) $c\neq 0$. Then $xy=c$ implies neither $x=0$ nor $y=0$. Therefore $y=c/x$ makes total sense. So in the first equation we get

$$x+y=b$$ $$x+c/x=b$$ $$x^2 + c=bx$$ $$x^2-bx+c=0$$

So all that is left is to solve this equation, which can have at most two solutions depending on $b,c$. Can you complete the calculations?