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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

  • x+x*sqrt((x- one)^ two)
  • x plus x multiply by square root of ((x minus 1) squared )
  • x plus x multiply by square root of ((x minus one) to the power of two)
  • x+x*√((x-1)^2)
  • x+x*sqrt((x-1)2)
  • x+x*sqrtx-12
  • x+x*sqrt((x-1)²)
  • x+x*sqrt((x-1) to the power of 2)
  • x+xsqrt((x-1)^2)
  • x+xsqrt((x-1)2)
  • x+xsqrtx-12
  • x+xsqrtx-1^2

  • Similar expressions

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

Graphing y = x+x*sqrt((x-1)^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                __________
               /        2 
f(x) = x + x*\/  (x - 1)
f(x)=x(x−1)2+xf{\left(x \right)} = x \sqrt{\left(x - 1\right)^{2}} + x 
f = x*sqrt((x - 1)^2) + x
The graph of the function
02468-8-6-4-2-1010-250250
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=0x \sqrt{\left(x - 1\right)^{2}} + x = 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*sqrt((x - 1)^2).
0(−1)20 \sqrt{\left(-1\right)^{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
x(x−1)∣x−1∣(x−1)2+(x−1)2+1=0\frac{x \left(x - 1\right) \left|{x - 1}\right|}{\left(x - 1\right)^{2}} + \sqrt{\left(x - 1\right)^{2}} + 1 = 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
xsign⁡(x−1)−x∣x−1∣x−1+2∣x−1∣x−1=0\frac{x \operatorname{sign}{\left(x - 1 \right)} - \frac{x \left|{x - 1}\right|}{x - 1} + 2 \left|{x - 1}\right|}{x - 1} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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(x \sqrt{\left(x - 1\right)^{2}} + x\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(x \sqrt{\left(x - 1\right)^{2}} + x\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 + x*sqrt((x - 1)^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x(x−1)2+xx)=∞\lim_{x \to -\infty}\left(\frac{x \sqrt{\left(x - 1\right)^{2}} + x}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞(x(x−1)2+xx)=∞\lim_{x \to \infty}\left(\frac{x \sqrt{\left(x - 1\right)^{2}} + x}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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∣−xx \sqrt{\left(x - 1\right)^{2}} + x = - x \left|{x + 1}\right| - x
- No
x(x−1)2+x=x∣x+1∣+xx \sqrt{\left(x - 1\right)^{2}} + x = x \left|{x + 1}\right| + x
- No
so, the function
not is
neither even, nor odd
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