# What is the graphic of f(x) = sqrt(x+sqrt(x+sqrt(x+sqrt(x+...)))) for x ge 0?

## What is the graphic of $f \left(x\right) = \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}$ for $x \ge 0$?

May 20, 2016

This is the continued-surd model for the equation of part of a parabola, in the first quadrant. Not in the graph, the vertex is at #(-1/4, 1.2) and the focus is at (0, 1/2).

#### Explanation:

As of now, $y = f \left(x\right) \ge 0$. Then $y = + \sqrt{x + y} , x \ge 0$.. Rationalizing,

${y}^{2} = x + y .$. Remodeling,

${\left(y - \frac{1}{2}\right)}^{2} = \left(x + \frac{1}{4}\right)$.

The graph is the part of a parabola that has vertex at $\left(- \frac{1}{4} , \frac{1}{2}\right)$

and latus rectum 4a = 1.. The focus is at $\left(0 , \frac{1}{2}\right)$.

As $x \mathmr{and} y \ge 0$, the graph is the part of the parabola in the 1st

quadrant, wherein $y > 1$..

I think it is better to restrict x as > 0, to avoid (0, 1) of the parabola.

Unlike parabola y, our y is single-valued, with $f \left(x\right) \in \left(1 , \infty\right)$.

$f \left(4\right) = \frac{1 + \sqrt{17}}{2} = 2.56$ nearly. See this plot, in the graph.

graph{(x+y-y^2)((x-4)^2+(y-2.56)^2-.001)=0[0.1 5 1 5] }

I make it for another g in continued-surd $y = \sqrt{g \left(x\right) + y}$ .

Let g(x) = ln x. Then $y = \sqrt{\ln x + \sqrt{\ln x + \sqrt{\ln x + \ldots}}}$.

Here, $x \ge {e}^{- 0.25} = 0.7788 \ldots$.Observe that y is single valued for

$x \ge 1$. See the plot is (1, 1).
graph{((ln x+y)^0.5-y)((x-1)^2+(y-1)^2-.01)=0[0..779 1 0.1 1] }