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Plotting a function graph/ (3*|x|-11)/(x-2)

Graphing y = (3*|x|-11)/(x-2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       3*|x| - 11
f(x) = ----------
         x - 2
f(x)=3∣x∣−11x−2f{\left(x \right)} = \frac{3 \left|{x}\right| - 11}{x - 2} 
f = (3*|x| - 11)/(x - 2)
The graph of the function
02468-8-6-4-2-1010-250250
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = 2
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:
3∣x∣−11x−2=0\frac{3 \left|{x}\right| - 11}{x - 2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−113x_{1} = - \frac{11}{3}
x2=113x_{2} = \frac{11}{3}
Numerical solution
x1=3.66666666666667x_{1} = 3.66666666666667
x2=−3.66666666666667x_{2} = -3.66666666666667
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*|x| - 11)/(x - 2).
−11+3∣0∣−2\frac{-11 + 3 \left|{0}\right|}{-2}
The result:
f(0)=112f{\left(0 \right)} = \frac{11}{2}
The point:
(0, 11/2)
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
3sign⁡(x)x−2−3∣x∣−11(x−2)2=0\frac{3 \operatorname{sign}{\left(x \right)}}{x - 2} - \frac{3 \left|{x}\right| - 11}{\left(x - 2\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(3δ(x)−3sign⁡(x)x−2+3∣x∣−11(x−2)2)x−2=0\frac{2 \left(3 \delta\left(x\right) - \frac{3 \operatorname{sign}{\left(x \right)}}{x - 2} + \frac{3 \left|{x}\right| - 11}{\left(x - 2\right)^{2}}\right)}{x - 2} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=2x_{1} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3∣x∣−11x−2)=−3\lim_{x \to -\infty}\left(\frac{3 \left|{x}\right| - 11}{x - 2}\right) = -3
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−3y = -3
lim⁡x→∞(3∣x∣−11x−2)=3\lim_{x \to \infty}\left(\frac{3 \left|{x}\right| - 11}{x - 2}\right) = 3
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=3y = 3
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (3*|x| - 11)/(x - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3∣x∣−11x(x−2))=0\lim_{x \to -\infty}\left(\frac{3 \left|{x}\right| - 11}{x \left(x - 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3∣x∣−11x(x−2))=0\lim_{x \to \infty}\left(\frac{3 \left|{x}\right| - 11}{x \left(x - 2\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:
3∣x∣−11x−2=3∣x∣−11−x−2\frac{3 \left|{x}\right| - 11}{x - 2} = \frac{3 \left|{x}\right| - 11}{- x - 2}
- No
3∣x∣−11x−2=−3∣x∣−11−x−2\frac{3 \left|{x}\right| - 11}{x - 2} = - \frac{3 \left|{x}\right| - 11}{- x - 2}
- No
so, the function
not is
neither even, nor odd
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