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Plotting a function graph/ 1/exp(x)*(x+1)

Graphing y = 1/exp(x)*(x+1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       x + 1
f(x) = -----
          x 
         e
f(x)=x+1exf{\left(x \right)} = \frac{x + 1}{e^{x}} 
f = (x + 1)/exp(x)
The graph of the function
02468-8-6-4-2-1010-200000200000
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:
x+1ex=0\frac{x + 1}{e^{x}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=75.51136695866x_{1} = 75.51136695866
x2=93.4384664647568x_{2} = 93.4384664647568
x3=97.4264551520843x_{3} = 97.4264551520843
x4=53.6879649775293x_{4} = 53.6879649775293
x5=77.5012725708786x_{5} = 77.5012725708786
x6=85.466430197318x_{6} = 85.466430197318
x7=121.37285448328x_{7} = 121.37285448328
x8=79.4917816149558x_{8} = 79.4917816149558
x9=117.380091923383x_{9} = 117.380091923383
x10=55.664342946604x_{10} = 55.664342946604
x11=99.4208627025251x_{11} = 99.4208627025251
x12=105.405524706139x_{12} = 105.405524706139
x13=113.387900375534x_{13} = 113.387900375534
x14=73.5221246603965x_{14} = 73.5221246603965
x15=45.8119589630405x_{15} = 45.8119589630405
x16=71.5336138177003x_{16} = 71.5336138177003
x17=32.3071598061728x_{17} = 32.3071598061728
x18=89.4517230466241x_{18} = 89.4517230466241
x19=87.4588807455217x_{19} = 87.4588807455217
x20=69.545912319012x_{20} = 69.545912319012
x21=39.9557499214057x_{21} = 39.9557499214057
x22=47.7754697845928x_{22} = 47.7754697845928
x23=−1x_{23} = -1
x24=36.0970717014418x_{24} = 36.0970717014418
x25=57.642856145511x_{25} = 57.642856145511
x26=119.376405823956x_{26} = 119.376405823956
x27=59.6232240789579x_{27} = 59.6232240789579
x28=81.4828412467504x_{28} = 81.4828412467504
x29=107.400841299949x_{29} = 107.400841299949
x30=38.020216210141x_{30} = 38.020216210141
x31=41.9008089996782x_{31} = 41.9008089996782
x32=111.392040334004x_{32} = 111.392040334004
x33=95.432316424891x_{33} = 95.432316424891
x34=115.383920620405x_{34} = 115.383920620405
x35=91.444927247289x_{35} = 91.444927247289
x36=65.5733090128955x_{36} = 65.5733090128955
x37=34.1905363866884x_{37} = 34.1905363866884
x38=67.5591096232555x_{38} = 67.5591096232555
x39=43.853370487631x_{39} = 43.853370487631
x40=51.714063380457x_{40} = 51.714063380457
x41=83.4744046501982x_{41} = 83.4744046501982
x42=49.7430576092052x_{42} = 49.7430576092052
x43=101.415520933891x_{43} = 101.415520933891
x44=103.410413305772x_{44} = 103.410413305772
x45=61.6052138551392x_{45} = 61.6052138551392
x46=63.5886304003902x_{46} = 63.5886304003902
x47=109.396350396671x_{47} = 109.396350396671
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x + 1)/exp(x).
1e0\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
−(x+1)e−x+1ex=0- \left(x + 1\right) e^{- x} + \frac{1}{e^{x}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(0, 1)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \infty\right)
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
(x−1)e−x=0\left(x - 1\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 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
[1,∞)\left[1, \infty\right)
Convex at the intervals
(−∞,1]\left(-\infty, 1\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x+1ex)=−∞\lim_{x \to -\infty}\left(\frac{x + 1}{e^{x}}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x+1ex)=0\lim_{x \to \infty}\left(\frac{x + 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 (x + 1)/exp(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x+1)e−xx)=∞\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) e^{- x}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((x+1)e−xx)=0\lim_{x \to \infty}\left(\frac{\left(x + 1\right) 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:
x+1ex=(1−x)ex\frac{x + 1}{e^{x}} = \left(1 - x\right) e^{x}
- No
x+1ex=−(1−x)ex\frac{x + 1}{e^{x}} = - \left(1 - x\right) e^{x}
- No
so, the function
not is
neither even, nor odd
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