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  • How to use it?

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

  • Identical expressions

  • ((x- one)/(x- one))^ two
  • ((x minus 1) divide by (x minus 1)) squared
  • ((x minus one) divide by (x minus one)) to the power of two
  • ((x-1)/(x-1))2
  • x-1/x-12
  • ((x-1)/(x-1))²
  • ((x-1)/(x-1)) to the power of 2
  • x-1/x-1^2
  • ((x-1) divide by (x-1))^2

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
              2
       /x - 1\ 
f(x) = |-----| 
       \x - 1/
f(x)=(x−1x−1)2f{\left(x \right)} = \left(\frac{x - 1}{x - 1}\right)^{2} 
f = ((x - 1)/(x - 1))^2
The graph of the function
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
The domain of the function
The points at which the function is not precisely defined:
x1=1x_{1} = 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−1x−1)2=0\left(\frac{x - 1}{x - 1}\right)^{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 ((x - 1)/(x - 1))^2.
(−1−1)2\left(- \frac{1}{-1}\right)^{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
0=00 = 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
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=1x_{1} = 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−1x−1)2=1\lim_{x \to -\infty} \left(\frac{x - 1}{x - 1}\right)^{2} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(x−1x−1)2=1\lim_{x \to \infty} \left(\frac{x - 1}{x - 1}\right)^{2} = 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 ((x - 1)/(x - 1))^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞1x=0\lim_{x \to -\infty} \frac{1}{x} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞1x=0\lim_{x \to \infty} \frac{1}{x} = 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−1x−1)2=1\left(\frac{x - 1}{x - 1}\right)^{2} = 1
- No
(x−1x−1)2=−1\left(\frac{x - 1}{x - 1}\right)^{2} = -1
- No
so, the function
not is
neither even, nor odd
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