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Plotting a function graph/ -ln1+x/1-x

Graphing y = -ln1+x/1-x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                 x    
f(x) = -log(1) + - - x
                 1
f(x)=−x+(x1−log⁡(1))f{\left(x \right)} = - x + \left(\frac{x}{1} - \log{\left(1 \right)}\right) 
f = -x + x/1 - log(1)
The graph of the function
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
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+(x1−log⁡(1))=0- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right) = 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 -log(1) + x/1 - x.
(−log⁡(1)+01)−0\left(- \log{\left(1 \right)} + \frac{0}{1}\right) - 0
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
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
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−x+(x1−log⁡(1)))=0\lim_{x \to -\infty}\left(- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right)\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(−x+(x1−log⁡(1)))=0\lim_{x \to \infty}\left(- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right)\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -log(1) + x/1 - x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−x+(x1−log⁡(1))x)=0\lim_{x \to -\infty}\left(\frac{- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right)}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−x+(x1−log⁡(1))x)=0\lim_{x \to \infty}\left(\frac{- x + \left(\frac{x}{1} - \log{\left(1 \right)}\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:
−x+(x1−log⁡(1))=−log⁡(1)- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right) = - \log{\left(1 \right)}
- No
−x+(x1−log⁡(1))=log⁡(1)- x + \left(\frac{x}{1} - \log{\left(1 \right)}\right) = \log{\left(1 \right)}
- No
so, the function
not is
neither even, nor odd
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