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Plotting a function graph/ ln(ln(x))

Graphing y = ln(ln(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = log(log(x))
f(x)=log⁡(log⁡(x))f{\left(x \right)} = \log{\left(\log{\left(x \right)} \right)} 
f = log(log(x))
The graph of the function
5501015202530354045-1010
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
log⁡(log⁡(x))=0\log{\left(\log{\left(x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=ex_{1} = e
Numerical solution
x1=2.71828182845905x_{1} = 2.71828182845905
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(log(x)).
log⁡(log⁡(0))\log{\left(\log{\left(0 \right)} \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
1xlog⁡(x)=0\frac{1}{x \log{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−1+1log⁡(x)x2log⁡(x)=0- \frac{1 + \frac{1}{\log{\left(x \right)}}}{x^{2} \log{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=e−1x_{1} = e^{-1}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,e−1]\left(-\infty, e^{-1}\right]
Convex at the intervals
[e−1,∞)\left[e^{-1}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(log⁡(x))=∞\lim_{x \to -\infty} \log{\left(\log{\left(x \right)} \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞log⁡(log⁡(x))=∞\lim_{x \to \infty} \log{\left(\log{\left(x \right)} \right)} = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(log(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(log⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(log⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
log⁡(log⁡(x))=log⁡(log⁡(−x))\log{\left(\log{\left(x \right)} \right)} = \log{\left(\log{\left(- x \right)} \right)}
- No
log⁡(log⁡(x))=−log⁡(log⁡(−x))\log{\left(\log{\left(x \right)} \right)} = - \log{\left(\log{\left(- x \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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