This visualization shows how the number line (in red) is mapped by the function

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

to produce the points shown in green.

The orange lines indicate the fixed points of $f$, which satisfy

$$ x = f(x). $$

We can find these fixed points by solving the equation:

$$ x = 1 + \frac{1}{x} $$

Rearranging,

$$ x - 1 - \frac{1}{x} = 0 $$

Multiplying both sides by $x$ to clear the denominator:

$$ x^2 - x - 1 = 0 $$

The solutions to this quadratic are the fixed points of the function. Fixed points are where applying the function $f$ does not change the point, i.e., $x = f(x)$. In this case, the fixed points are the golden ratio and its conjugate.