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  • Graphing y =:
  • (x+1)^3/(x-1)^2
  • 9x-x^3
  • 4x^2-x^2
  • 3x^2+2x-1

  • Identical expressions

  • exp(x/(x^ two - one))
  • exponent of (x divide by (x squared minus 1))
  • exponent of (x divide by (x to the power of two minus one))
  • exp(x/(x2-1))
  • expx/x2-1
  • exp(x/(x²-1))
  • exp(x/(x to the power of 2-1))
  • expx/x^2-1
  • exp(x divide by (x^2-1))

  • Similar expressions

  • exp(x/(x^2+1))
Plotting a function graph/ exp(x/(x^2-1))

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          x   
        ------
         2    
        x  - 1
f(x) = e
f(x)=exx2−1f{\left(x \right)} = e^{\frac{x}{x^{2} - 1}} 
f = exp(x/(x^2 - 1))
The graph of the function
02468-8-6-4-2-1010050000
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -1
x2=1x_{2} = 1
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:
exx2−1=0e^{\frac{x}{x^{2} - 1}} = 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(x/(x^2 - 1)).
e0−1+02e^{\frac{0}{-1 + 0^{2}}}
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
(−2x2(x2−1)2+1x2−1)exx2−1=0\left(- \frac{2 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{1}{x^{2} - 1}\right) e^{\frac{x}{x^{2} - 1}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞exx2−1=1\lim_{x \to -\infty} e^{\frac{x}{x^{2} - 1}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞exx2−1=1\lim_{x \to \infty} e^{\frac{x}{x^{2} - 1}} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of exp(x/(x^2 - 1)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(exx2−1x)=0\lim_{x \to -\infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(exx2−1x)=0\lim_{x \to \infty}\left(\frac{e^{\frac{x}{x^{2} - 1}}}{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:
exx2−1=e−xx2−1e^{\frac{x}{x^{2} - 1}} = e^{- \frac{x}{x^{2} - 1}}
- No
exx2−1=−e−xx2−1e^{\frac{x}{x^{2} - 1}} = - e^{- \frac{x}{x^{2} - 1}}
- No
so, the function
not is
neither even, nor odd
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