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exp^(2-x)/(2-x)*x

  • How to use it?

  • Graphing y =:
  • (x+2)^2/(x+1)
  • x^2-3x+3
  • -x^2-3
  • -x^2+2x+15

  • Identical expressions

  • exp^(two -x)/(two -x)*x
  • exponent of to the power of (2 minus x) divide by (2 minus x) multiply by x
  • exponent of to the power of (two minus x) divide by (two minus x) multiply by x
  • exp(2-x)/(2-x)*x
  • exp2-x/2-x*x
  • exp^(2-x)/(2-x)x
  • exp(2-x)/(2-x)x
  • exp2-x/2-xx
  • exp^2-x/2-xx
  • exp^(2-x) divide by (2-x)*x

  • Similar expressions

  • exp^(2+x)/(2-x)*x
  • exp^(2-x)/(2+x)*x
Plotting a function graph/ exp^(2-x)/(2-x)*x

Graphing y = exp^(2-x)/(2-x)*x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        2 - x  
       e     *x
f(x) = --------
        2 - x
f(x)=xe2−x2−xf{\left(x \right)} = \frac{x e^{2 - x}}{2 - x} 
f = x*E^(2 - x)/(2 - x)
The graph of the function
0102030405060708090100110120-10-200000100000
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = 2
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:
xe2−x2−x=0\frac{x e^{2 - x}}{2 - x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=33.0571776157327x_{1} = 33.0571776157327
x2=121.178102581297x_{2} = 121.178102581297
x3=87.1740763981455x_{3} = 87.1740763981455
x4=51.1521535131471x_{4} = 51.1521535131471
x5=91.1748368518887x_{5} = 91.1748368518887
x6=69.1683555957443x_{6} = 69.1683555957443
x7=49.1486096800712x_{7} = 49.1486096800712
x8=55.15769628377x_{8} = 55.15769628377
x9=89.1744717733504x_{9} = 89.1744717733504
x10=57.1598899821613x_{10} = 57.1598899821613
x11=115.177672108263x_{11} = 115.177672108263
x12=111.177340056896x_{12} = 111.177340056896
x13=41.124918517446x_{13} = 41.124918517446
x14=93.175174668055x_{14} = 93.175174668055
x15=107.176964578548x_{15} = 107.176964578548
x16=53.1551454444043x_{16} = 53.1551454444043
x17=99.1760496128704x_{17} = 99.1760496128704
x18=101.176302029271x_{18} = 101.176302029271
x19=71.1692685841321x_{19} = 71.1692685841321
x20=119.177967199791x_{20} = 119.177967199791
x21=43.132871687516x_{21} = 43.132871687516
x22=63.1649073358433x_{22} = 63.1649073358433
x23=95.1754878837313x_{23} = 95.1754878837313
x24=79.1721141752132x_{24} = 79.1721141752132
x25=83.1731803564069x_{25} = 83.1731803564069
x26=77.1715033698701x_{26} = 77.1715033698701
x27=65.1661941485493x_{27} = 65.1661941485493
x28=117.177823918444x_{28} = 117.177823918444
x29=85.1736472548213x_{29} = 85.1736472548213
x30=73.1700898713663x_{30} = 73.1700898713663
x31=61.1634503178614x_{31} = 61.1634503178614
x32=45.1392149059937x_{32} = 45.1392149059937
x33=81.1726711050428x_{33} = 81.1726711050428
x34=103.176537720436x_{34} = 103.176537720436
x35=109.177158203895x_{35} = 109.177158203895
x36=35.0830976103373x_{36} = 35.0830976103373
x37=75.1708314419527x_{37} = 75.1708314419527
x38=67.1673365541753x_{38} = 67.1673365541753
x39=39.1147302712608x_{39} = 39.1147302712608
x40=113.177511076155x_{40} = 113.177511076155
x41=105.176758137916x_{41} = 105.176758137916
x42=31.0180079017645x_{42} = 31.0180079017645
x43=0x_{43} = 0
x44=37.1013285290104x_{44} = 37.1013285290104
x45=97.1757788421662x_{45} = 97.1757788421662
x46=47.1443649826676x_{46} = 47.1443649826676
x47=59.1617911706003x_{47} = 59.1617911706003
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to E^(2 - x)*x/(2 - x).
e2−0⋅12−0⋅0e^{2 - 0} \cdot \frac{1}{2 - 0} \cdot 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
−xe2−x2−x+xe2−x(2−x)2+e2−x2−x=0- \frac{x e^{2 - x}}{2 - x} + \frac{x e^{2 - x}}{\left(2 - x\right)^{2}} + \frac{e^{2 - x}}{2 - 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
(−x−2xx−2−2x(x−2)2+2+2x−2)e2−xx−2=0\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2} = 0
Solve this equation
The roots of this equation
x1=−26+63333+8326+6333+43x_{1} = - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=2x_{1} = 2

lim⁡x→2−((−x−2xx−2−2x(x−2)2+2+2x−2)e2−xx−2)=∞\lim_{x \to 2^-}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = \infty
Let's take the limit
lim⁡x→2+((−x−2xx−2−2x(x−2)2+2+2x−2)e2−xx−2)=−∞\lim_{x \to 2^+}\left(\frac{\left(- x - \frac{2 x}{x - 2} - \frac{2 x}{\left(x - 2\right)^{2}} + 2 + \frac{2}{x - 2}\right) e^{2 - x}}{x - 2}\right) = -\infty
Let's take the limit
- the limits are not equal, so
x1=2x_{1} = 2
- is an inflection point

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
[−26+63333+8326+6333+43,∞)\left[- \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}, \infty\right)
Convex at the intervals
(−∞,−26+63333+8326+6333+43]\left(-\infty, - \frac{\sqrt[3]{26 + 6 \sqrt{33}}}{3} + \frac{8}{3 \sqrt[3]{26 + 6 \sqrt{33}}} + \frac{4}{3}\right]
Vertical asymptotes
Have:
x1=2x_{1} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xe2−x2−x)=−∞\lim_{x \to -\infty}\left(\frac{x e^{2 - x}}{2 - x}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(xe2−x2−x)=0\lim_{x \to \infty}\left(\frac{x e^{2 - x}}{2 - 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 E^(2 - x)*x/(2 - x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(e2−x2−x)=∞\lim_{x \to -\infty}\left(\frac{e^{2 - x}}{2 - x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(e2−x2−x)=0\lim_{x \to \infty}\left(\frac{e^{2 - x}}{2 - 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:
xe2−x2−x=−xex+2x+2\frac{x e^{2 - x}}{2 - x} = - \frac{x e^{x + 2}}{x + 2}
- No
xe2−x2−x=xex+2x+2\frac{x e^{2 - x}}{2 - x} = \frac{x e^{x + 2}}{x + 2}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = exp^(2-x)/(2-x)*x
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