Graphing y = e^5^x

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        / x\
        \5 /
f(x) = e

$$f{\left(x \right)} = e^{5^{x}}$$

f = E^(5^x)

The graph of the function

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:
$$e^{5^{x}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to E^(5^x).
$$e^{5^{0}}$$
The result:
$$f{\left(0 \right)} = e$$
The point:

(0, E)

Extrema of the function

In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$5^{x} e^{5^{x}} \log{\left(5 \right)} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$5^{x} \left(5^{x} + 1\right) e^{5^{x}} \log{\left(5 \right)}^{2} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} e^{5^{x}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} e^{5^{x}} = \infty$$
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 E^(5^x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{5^{x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{5^{x}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist

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:
$$e^{5^{x}} = e^{5^{- x}}$$
- No
$$e^{5^{x}} = - e^{5^{- x}}$$
- No
so, the function
not is
neither even, nor odd

The graph

Other calculators

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Identical expressions

Graphing y = e^5^x

The graph:

Intersection points:

Piecewise:

The solution