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  • How to use it?

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

  • Identical expressions

  • xsqrt(x- one)^ two /|x|
  • x square root of (x minus 1) squared divide by module of x|
  • x square root of (x minus one) to the power of two divide by module of x|
  • x√(x-1)^2/|x|
  • xsqrt(x-1)2/|x|
  • xsqrtx-12/|x|
  • xsqrt(x-1)²/|x|
  • xsqrt(x-1) to the power of 2/|x|
  • xsqrtx-1^2/|x|
  • xsqrt(x-1)^2 divide by |x|

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                  2
           _______ 
       x*\/ x - 1  
f(x) = ------------
           |x|
f(x)=x(x−1)2∣x∣f{\left(x \right)} = \frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} 
f = (x*(sqrt(x - 1))^2)/|x|
The graph of the function
02468-8-6-4-2-101020-10
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:
x(x−1)2∣x∣=0\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1x_{1} = 1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x*(sqrt(x - 1))^2)/|x|.
0(−1)2∣0∣\frac{0 \left(\sqrt{-1}\right)^{2}}{\left|{0}\right|}
The result:
f(0)=NaNf{\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
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
x+(x−1)2∣x∣−(x−1)sign⁡(x)x=0\frac{x + \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} - \frac{\left(x - 1\right) \operatorname{sign}{\left(x \right)}}{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
2(1∣x∣−(x−1)(δ(x)−sign⁡(x)x)x−(2x−1)sign⁡(x)x2)=02 \left(\frac{1}{\left|{x}\right|} - \frac{\left(x - 1\right) \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)}}{x}\right)}{x} - \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)}}{x^{2}}\right) = 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
lim⁡x→−∞(x(x−1)2∣x∣)=∞\lim_{x \to -\infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x(x−1)2∣x∣)=∞\lim_{x \to \infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x*(sqrt(x - 1))^2)/|x|, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x−1∣x∣)=−1\lim_{x \to -\infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the left:
y=−xy = - x
lim⁡x→∞(x−1∣x∣)=1\lim_{x \to \infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
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(x−1)2∣x∣=−x(−x−1)∣x∣\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = - \frac{x \left(- x - 1\right)}{\left|{x}\right|}
- No
x(x−1)2∣x∣=x(−x−1)∣x∣\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = \frac{x \left(- x - 1\right)}{\left|{x}\right|}
- No
so, the function
not is
neither even, nor odd
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