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Plotting a function graph/ x-2/(x-3)*(x+1)

Graphing y = x-2/(x-3)*(x+1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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             2          
f(x) = x - -----*(x + 1)
           x - 3
f(x)=x−2x−3(x+1)f{\left(x \right)} = x - \frac{2}{x - 3} \left(x + 1\right) 
f = x - 2/(x - 3)*(x + 1)
The graph of the function
02468-8-6-4-2-1010-500500
The domain of the function
The points at which the function is not precisely defined:
x1=3x_{1} = 3
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−2x−3(x+1)=0x - \frac{2}{x - 3} \left(x + 1\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=52−332x_{1} = \frac{5}{2} - \frac{\sqrt{33}}{2}
x2=52+332x_{2} = \frac{5}{2} + \frac{\sqrt{33}}{2}
Numerical solution
x1=−0.372281323269014x_{1} = -0.372281323269014
x2=5.37228132326901x_{2} = 5.37228132326901
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 - 3)*(x + 1).
−2−3- \frac{2}{-3}
The result:
f(0)=23f{\left(0 \right)} = \frac{2}{3}
The point:
(0, 2/3)
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
1−2x−3+2(x+1)(x−3)2=01 - \frac{2}{x - 3} + \frac{2 \left(x + 1\right)}{\left(x - 3\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
4(1−x+1x−3)(x−3)2=0\frac{4 \left(1 - \frac{x + 1}{x - 3}\right)}{\left(x - 3\right)^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=3x_{1} = 3
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x−2x−3(x+1))=−∞\lim_{x \to -\infty}\left(x - \frac{2}{x - 3} \left(x + 1\right)\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x−2x−3(x+1))=∞\lim_{x \to \infty}\left(x - \frac{2}{x - 3} \left(x + 1\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 x - 2/(x - 3)*(x + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x−2x−3(x+1)x)=1\lim_{x \to -\infty}\left(\frac{x - \frac{2}{x - 3} \left(x + 1\right)}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the left:
y=xy = x
lim⁡x→∞(x−2x−3(x+1)x)=1\lim_{x \to \infty}\left(\frac{x - \frac{2}{x - 3} \left(x + 1\right)}{x}\right) = 1
Let's take the limit
so,
inclined asymptote equation on the right:
y=xy = x
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−2x−3(x+1)=−x−2(1−x)−x−3x - \frac{2}{x - 3} \left(x + 1\right) = - x - \frac{2 \left(1 - x\right)}{- x - 3}
- No
x−2x−3(x+1)=x+2(1−x)−x−3x - \frac{2}{x - 3} \left(x + 1\right) = x + \frac{2 \left(1 - x\right)}{- x - 3}
- No
so, the function
not is
neither even, nor odd
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