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Plotting a function graph/ x/(2*x-1)

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          x   
f(x) = -------
       2*x - 1
f(x)=x2x−1f{\left(x \right)} = \frac{x}{2 x - 1} 
f = x/(2*x - 1)
The graph of the function
02468-8-6-4-2-1010-1010
The domain of the function
The points at which the function is not precisely defined:
x1=0.5x_{1} = 0.5
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:
x2x−1=0\frac{x}{2 x - 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/(2*x - 1).
0−1+0⋅2\frac{0}{-1 + 0 \cdot 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(2x−1)2+12x−1=0- \frac{2 x}{\left(2 x - 1\right)^{2}} + \frac{1}{2 x - 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
4(2x2x−1−1)(2x−1)2=0\frac{4 \left(\frac{2 x}{2 x - 1} - 1\right)}{\left(2 x - 1\right)^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0.5x_{1} = 0.5
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x2x−1)=12\lim_{x \to -\infty}\left(\frac{x}{2 x - 1}\right) = \frac{1}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=12y = \frac{1}{2}
lim⁡x→∞(x2x−1)=12\lim_{x \to \infty}\left(\frac{x}{2 x - 1}\right) = \frac{1}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=12y = \frac{1}{2}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(2*x - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞12x−1=0\lim_{x \to -\infty} \frac{1}{2 x - 1} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞12x−1=0\lim_{x \to \infty} \frac{1}{2 x - 1} = 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:
x2x−1=−x−2x−1\frac{x}{2 x - 1} = - \frac{x}{- 2 x - 1}
- No
x2x−1=x−2x−1\frac{x}{2 x - 1} = \frac{x}{- 2 x - 1}
- No
so, the function
not is
neither even, nor odd
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