Reciprocal of Quadratic Equation

Question

How can we prove there are infinitely many solutions to $\frac{1}{x^{2}-2x+3}=y$ by only staying at Further maths at High School level? Will the graph ever go below the x-axis or will stay on it. Graph of the equation above

Answer 1

The curve in the graph gets infinitely close to the $x$-axis as $x\to\pm\infty$. However, it always stay above the axis, which can be seen analytically from,

$$\frac{1}{x^{2}-2x+3}=\frac{1}{(x-1)^{2}+2}> 0$$

Thus, no solutions exist.

Answer 2

The denominator $(x-1)^2+2$ is positive so reciprocal can never intersect x-axis. So no real roots but complex conjugate roots with real part near $\approx 1.$