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(x) = sqrt(x+sqrt(x+sqrt(x+sqrt(x+...)))) for x ge 0?

1 Answer
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(x)>=0. Then y =+sqrt(x+y), x>=0.. Rationalizing,

y^2=x+y.. Remodeling,

(y-1/2)^2=(x+1/4).

The graph is the part of a parabola that has vertex at (-1/4, 1/2)

and latus rectum 4a = 1.. The focus is at (0, 1/2).

As x and y >= 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(x) in (1, oo).

f(4) = (1 + sqrt17)/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(x)+y) .

Let g(x) = ln x. Then y = sqrt(ln x + sqrt(ln x + sqrt(ln x +...))).

Here, x >= e^(-0.25) = 0.7788....Observe that y is single valued for

x >=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] }