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

  • Identical expressions

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

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
        |x|  
f(x) = ------
        2    
       x  - 1
f(x)=∣x∣x2−1f{\left(x \right)} = \frac{\left|{x}\right|}{x^{2} - 1} 
f = |x|/(x^2 - 1)
The graph of the function
02468-8-6-4-2-1010-2020
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:
∣x∣x2−1=0\frac{\left|{x}\right|}{x^{2} - 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to |x|/(x^2 - 1).
∣0∣−1+02\frac{\left|{0}\right|}{-1 + 0^{2}}
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
−2x∣x∣(x2−1)2+sign⁡(x)x2−1=0- \frac{2 x \left|{x}\right|}{\left(x^{2} - 1\right)^{2}} + \frac{\operatorname{sign}{\left(x \right)}}{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→−∞(∣x∣x2−1)=0\lim_{x \to -\infty}\left(\frac{\left|{x}\right|}{x^{2} - 1}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(∣x∣x2−1)=0\lim_{x \to \infty}\left(\frac{\left|{x}\right|}{x^{2} - 1}\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|/(x^2 - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣x∣x(x2−1))=0\lim_{x \to -\infty}\left(\frac{\left|{x}\right|}{x \left(x^{2} - 1\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(∣x∣x(x2−1))=0\lim_{x \to \infty}\left(\frac{\left|{x}\right|}{x \left(x^{2} - 1\right)}\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:
∣x∣x2−1=∣x∣x2−1\frac{\left|{x}\right|}{x^{2} - 1} = \frac{\left|{x}\right|}{x^{2} - 1}
- Yes
∣x∣x2−1=−∣x∣x2−1\frac{\left|{x}\right|}{x^{2} - 1} = - \frac{\left|{x}\right|}{x^{2} - 1}
- No
so, the function
is
even
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