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Plotting a function graph/ (x+1)/(x-1)

Graphing y = (x+1)/(x-1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       x + 1
f(x) = -----
       x - 1
f(x)=x+1x−1f{\left(x \right)} = \frac{x + 1}{x - 1} 
f = (x + 1)/(x - 1)
The graph of the function
02468-8-6-4-2-1010-100100
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=0\frac{x + 1}{x - 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=−1x_{1} = -1
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).
1−1\frac{1}{-1}
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
1x−1−x+1(x−1)2=0\frac{1}{x - 1} - \frac{x + 1}{\left(x - 1\right)^{2}} = 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(−1+x+1x−1)(x−1)2=0\frac{2 \left(-1 + \frac{x + 1}{x - 1}\right)}{\left(x - 1\right)^{2}} = 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)=1\lim_{x \to -\infty}\left(\frac{x + 1}{x - 1}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(x+1x−1)=1\lim_{x \to \infty}\left(\frac{x + 1}{x - 1}\right) = 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), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x+1x(x−1))=0\lim_{x \to -\infty}\left(\frac{x + 1}{x \left(x - 1\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x+1x(x−1))=0\lim_{x \to \infty}\left(\frac{x + 1}{x \left(x - 1\right)}\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+1x−1=1−x−x−1\frac{x + 1}{x - 1} = \frac{1 - x}{- x - 1}
- No
x+1x−1=−1−x−x−1\frac{x + 1}{x - 1} = - \frac{1 - x}{- x - 1}
- No
so, the function
not is
neither even, nor odd
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