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  • Graphing y =:
  • 3x^4-4x^3
  • 3/2x^2-x^3
  • 3x^2-12x
  • -2x^2+x+1

  • Identical expressions

  • two *log(one / two (x- one))
  • 2 multiply by logarithm of (1 divide by 2(x minus 1))
  • two multiply by logarithm of (one divide by two (x minus one))
  • 2log(1/2(x-1))
  • 2log1/2x-1
  • 2*log(1 divide by 2(x-1))

  • Similar expressions

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

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /x - 1\
f(x) = 2*log|-----|
            \  2  /
f(x)=2log⁡(x−12)f{\left(x \right)} = 2 \log{\left(\frac{x - 1}{2} \right)} 
f = 2*log((x - 1)/2)
The graph of the function
02468-8-6-4-2-1010-1010
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:
2log⁡(x−12)=02 \log{\left(\frac{x - 1}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3x_{1} = 3
Numerical solution
x1=3x_{1} = 3
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*log((x - 1)/2).
2log⁡(−12)2 \log{\left(\frac{-1}{2} \right)}
The result:
f(0)=−2log⁡(2)+2iπf{\left(0 \right)} = - 2 \log{\left(2 \right)} + 2 i \pi
The point:
(0, -2*log(2) + 2*pi*i)
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−1=0\frac{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
−2(x−1)2=0- \frac{2}{\left(x - 1\right)^{2}} = 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→−∞(2log⁡(x−12))=∞\lim_{x \to -\infty}\left(2 \log{\left(\frac{x - 1}{2} \right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(2log⁡(x−12))=∞\lim_{x \to \infty}\left(2 \log{\left(\frac{x - 1}{2} \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 2*log((x - 1)/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2log⁡(x−12)x)=0\lim_{x \to -\infty}\left(\frac{2 \log{\left(\frac{x - 1}{2} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2log⁡(x−12)x)=0\lim_{x \to \infty}\left(\frac{2 \log{\left(\frac{x - 1}{2} \right)}}{x}\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:
2log⁡(x−12)=2log⁡(−x2−12)2 \log{\left(\frac{x - 1}{2} \right)} = 2 \log{\left(- \frac{x}{2} - \frac{1}{2} \right)}
- No
2log⁡(x−12)=−2log⁡(−x2−12)2 \log{\left(\frac{x - 1}{2} \right)} = - 2 \log{\left(- \frac{x}{2} - \frac{1}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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