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Plotting a function graph/ (x-1)e^-x

Graphing y = (x-1)e^-x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                -x
f(x) = (x - 1)*E
f(x)=e−x(x−1)f{\left(x \right)} = e^{- x} \left(x - 1\right) 
f = E^(-x)*(x - 1)
The graph of the function
02468-8-6-4-2-1010-250000250000
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−x(x−1)=0e^{- x} \left(x - 1\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=40.020216210141x_{1} = 40.020216210141
x2=97.432316424891x_{2} = 97.432316424891
x3=95.4384664647568x_{3} = 95.4384664647568
x4=101.420862702525x_{4} = 101.420862702525
x5=49.7754697845928x_{5} = 49.7754697845928
x6=65.5886304003902x_{6} = 65.5886304003902
x7=41.9557499214057x_{7} = 41.9557499214057
x8=32.4578471962376x_{8} = 32.4578471962376
x9=55.6879649775293x_{9} = 55.6879649775293
x10=91.4517230466241x_{10} = 91.4517230466241
x11=99.4264551520843x_{11} = 99.4264551520843
x12=75.5221246603965x_{12} = 75.5221246603965
x13=113.392040334004x_{13} = 113.392040334004
x14=119.380091923383x_{14} = 119.380091923383
x15=71.545912319012x_{15} = 71.545912319012
x16=59.642856145511x_{16} = 59.642856145511
x17=67.5733090128955x_{17} = 67.5733090128955
x18=34.3071598061728x_{18} = 34.3071598061728
x19=51.7430576092052x_{19} = 51.7430576092052
x20=87.466430197318x_{20} = 87.466430197318
x21=38.0970717014418x_{21} = 38.0970717014418
x22=117.383920620405x_{22} = 117.383920620405
x23=63.6052138551392x_{23} = 63.6052138551392
x24=53.714063380457x_{24} = 53.714063380457
x25=77.51136695866x_{25} = 77.51136695866
x26=107.405524706139x_{26} = 107.405524706139
x27=121.376405823956x_{27} = 121.376405823956
x28=83.4828412467504x_{28} = 83.4828412467504
x29=89.4588807455217x_{29} = 89.4588807455217
x30=105.410413305772x_{30} = 105.410413305772
x31=1x_{31} = 1
x32=57.664342946604x_{32} = 57.664342946604
x33=73.5336138177003x_{33} = 73.5336138177003
x34=61.6232240789579x_{34} = 61.6232240789579
x35=115.387900375534x_{35} = 115.387900375534
x36=103.415520933891x_{36} = 103.415520933891
x37=93.444927247289x_{37} = 93.444927247289
x38=111.396350396671x_{38} = 111.396350396671
x39=81.4917816149558x_{39} = 81.4917816149558
x40=109.400841299949x_{40} = 109.400841299949
x41=69.5591096232555x_{41} = 69.5591096232555
x42=36.1905363866884x_{42} = 36.1905363866884
x43=79.5012725708786x_{43} = 79.5012725708786
x44=85.4744046501982x_{44} = 85.4744046501982
x45=45.853370487631x_{45} = 45.853370487631
x46=47.8119589630405x_{46} = 47.8119589630405
x47=43.9008089996782x_{47} = 43.9008089996782
x48=32.2046743865559x_{48} = 32.2046743865559
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 1)*E^(-x).
−e−0- 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+e−x=0- \left(x - 1\right) e^{- x} + e^{- x} = 0
Solve this equation
The roots of this equation
x1=2x_{1} = 2
The values of the extrema at the points:
     -2 
(2, e  )


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=2x_{1} = 2
Decreasing at intervals
(−∞,2]\left(-\infty, 2\right]
Increasing at intervals
[2,∞)\left[2, \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−3)e−x=0\left(x - 3\right) e^{- x} = 0
Solve this equation
The roots of this equation
x1=3x_{1} = 3

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
[3,∞)\left[3, \infty\right)
Convex at the intervals
(−∞,3]\left(-\infty, 3\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(e−x(x−1))=−∞\lim_{x \to -\infty}\left(e^{- x} \left(x - 1\right)\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(e−x(x−1))=0\lim_{x \to \infty}\left(e^{- x} \left(x - 1\right)\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)*E^(-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:
e−x(x−1)=(−x−1)exe^{- x} \left(x - 1\right) = \left(- x - 1\right) e^{x}
- No
e−x(x−1)=−(−x−1)exe^{- x} \left(x - 1\right) = - \left(- x - 1\right) e^{x}
- No
so, the function
not is
neither even, nor odd
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