Graphing y = exp(y)

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        y
f(y) = e

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

f = exp(y)

The graph of the function

The points of intersection with the X-axis coordinate

Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
$$e^{y} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y

The points of intersection with the Y axis coordinate

The graph crosses Y axis when y equals 0:
substitute y = 0 to exp(y).
$$e^{0}$$
The result:
$$f{\left(0 \right)} = 1$$
The point:

(0, 1)

Extrema of the function

In order to find the extrema, we need to solve the equation
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$e^{y} = 0$$
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
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$e^{y} = 0$$
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 y->+oo and y->-oo
$$\lim_{y \to -\infty} e^{y} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{y \to \infty} e^{y} = \infty$$
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 exp(y), divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty}\left(\frac{e^{y}}{y}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(\frac{e^{y}}{y}\right) = \infty$$
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(-y) и f = -f(-y).
So, check:
$$e^{y} = e^{- y}$$
- No
$$e^{y} = - e^{- y}$$
- No
so, the function
not is
neither even, nor odd

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Identical expressions

Graphing y = exp(y)

The graph:

Intersection points:

Piecewise:

The solution