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1/e^x
Plotting a function graph/ 1/e^x

Graphing y = 1/e^x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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         1 
f(x) = 1*--
          x
         e
f(x)=1⋅1exf{\left(x \right)} = 1 \cdot \frac{1}{e^{x}} 
f = 1/E^x
The graph of the function
-2.0-1.5-1.0-0.52.00.00.51.01.5010
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:
1⋅1ex=01 \cdot \frac{1}{e^{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 1/E^x.
1⋅1e01 \cdot \frac{1}{e^{0}}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−e−x=0- e^{- x} = 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
e−x=0e^{- x} = 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→−∞(1⋅1ex)=∞\lim_{x \to -\infty}\left(1 \cdot \frac{1}{e^{x}}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(1⋅1ex)=0\lim_{x \to \infty}\left(1 \cdot \frac{1}{e^{x}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(e−xx)=−∞\lim_{x \to -\infty}\left(\frac{e^{- x}}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(e−xx)=0\lim_{x \to \infty}\left(\frac{e^{- x}}{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:
1⋅1ex=ex1 \cdot \frac{1}{e^{x}} = e^{x}
- No
1⋅1ex=−ex1 \cdot \frac{1}{e^{x}} = - e^{x}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 1/e^x
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