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Plotting a function graph/ exp(-sqrt(x))/(sqrt(x))

Graphing y = exp(-sqrt(x))/(sqrt(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           ___
        -\/ x 
       e      
f(x) = -------
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        \/ x
f(x)=e−xxf{\left(x \right)} = \frac{e^{- \sqrt{x}}}{\sqrt{x}} 
f = exp(-sqrt(x))/sqrt(x)
The graph of the function
1000020000300004000050000600007000080000900001000000.00000000000000001.0e-15
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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−xx=0\frac{e^{- \sqrt{x}}}{\sqrt{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 exp(-sqrt(x))/sqrt(x).
e−00\frac{e^{- \sqrt{0}}}{\sqrt{0}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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−x2xx−e−x2x32=0- \frac{e^{- \sqrt{x}}}{2 \sqrt{x} \sqrt{x}} - \frac{e^{- \sqrt{x}}}{2 x^{\frac{3}{2}}} = 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
(2x2+1x+1x32x+3x52)e−x4=0\frac{\left(\frac{2}{x^{2}} + \frac{\frac{1}{x} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{x}} + \frac{3}{x^{\frac{5}{2}}}\right) e^{- \sqrt{x}}}{4} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Limit on the left could not be calculated
lim⁡x→−∞(e−xx)\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x}}\right)
lim⁡x→∞(e−xx)=0\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{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 exp(-sqrt(x))/sqrt(x), divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated
lim⁡x→−∞(e−xxx)\lim_{x \to -\infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{x} x}\right)
lim⁡x→∞(e−xxx)=0\lim_{x \to \infty}\left(\frac{e^{- \sqrt{x}}}{\sqrt{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−xx=e−−x−x\frac{e^{- \sqrt{x}}}{\sqrt{x}} = \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}
- No
e−xx=−e−−x−x\frac{e^{- \sqrt{x}}}{\sqrt{x}} = - \frac{e^{- \sqrt{- x}}}{\sqrt{- x}}
- No
so, the function
not is
neither even, nor odd
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