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Plotting a function graph/ 1-exp^(-0.1*x)

Graphing y = 1-exp^(-0.1*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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            -x 
            ---
             10
f(x) = 1 - E
f(x)=1−e−x10f{\left(x \right)} = 1 - e^{- \frac{x}{10}} 
f = 1 - E^(-x/10)
The graph of the function
02468-8-6-4-2-10102.5-2.5
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−e−x10=01 - e^{- \frac{x}{10}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
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/10).
1−e−01 - e^{- 0}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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−x1010=0\frac{e^{- \frac{x}{10}}}{10} = 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−x10100=0- \frac{e^{- \frac{x}{10}}}{100} = 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−e−x10)=−∞\lim_{x \to -\infty}\left(1 - e^{- \frac{x}{10}}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(1−e−x10)=1\lim_{x \to \infty}\left(1 - e^{- \frac{x}{10}}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1 - E^(-x/10), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1−e−x10x)=∞\lim_{x \to -\infty}\left(\frac{1 - e^{- \frac{x}{10}}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(1−e−x10x)=0\lim_{x \to \infty}\left(\frac{1 - e^{- \frac{x}{10}}}{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−e−x10=1−ex101 - e^{- \frac{x}{10}} = 1 - e^{\frac{x}{10}}
- No
1−e−x10=ex10−11 - e^{- \frac{x}{10}} = e^{\frac{x}{10}} - 1
- No
so, the function
not is
neither even, nor odd
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