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  • Graphing y =:
  • 2x^4+8/3x^3
  • 2x^2-5x
  • 2x^2+6x
  • 1/3(x^3-14x^2+49x-36)

  • Identical expressions

  • exp^(one /(x^ two))
  • exponent of to the power of (1 divide by (x squared ))
  • exponent of to the power of (one divide by (x to the power of two))
  • exp(1/(x2))
  • exp1/x2
  • exp^(1/(x²))
  • exp to the power of (1/(x to the power of 2))
  • exp^1/x^2
  • exp^(1 divide by (x^2))
Plotting a function graph/ exp^(1/(x^2))

Graphing y = exp^(1/(x^2))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        1 
        --
         2
        x 
f(x) = E
$$f{\left(x \right)} = e^{\frac{1}{x^{2}}}$$ 
f = E^(1/(x^2))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
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:
$$e^{\frac{1}{x^{2}}} = 0$$
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 E^(1/(x^2)).
$$e^{\frac{1}{0^{2}}}$$
The result:
$$f{\left(0 \right)} = \text{NaN}$$
- the solutions of the equation d'not exist
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{2 e^{\frac{1}{x^{2}}}}{x^{3}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(3 + \frac{2}{x^{2}}\right) e^{\frac{1}{x^{2}}}}{x^{4}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} e^{\frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty} e^{\frac{1}{x^{2}}} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of E^(1/(x^2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{e^{\frac{1}{x^{2}}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{e^{\frac{1}{x^{2}}}}{x}\right) = 0$$
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:
$$e^{\frac{1}{x^{2}}} = e^{\frac{1}{x^{2}}}$$
- Yes
$$e^{\frac{1}{x^{2}}} = - e^{\frac{1}{x^{2}}}$$
- No
so, the function
is
even
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