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

Graphing y = (2lnx/x((x+3)/x))-3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       2*log(x)*(x + 3)    
f(x) = ---------------- - 3
             x*x
f(x)=2(x+3)log⁡(x)xx−3f{\left(x \right)} = \frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 
f = -1*3 + 2*(x + 3)*log(x)/(x*x)
The graph of the function
02468-8-6-4-2-1010-1000010000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
2(x+3)log⁡(x)xx−3=0\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 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 2*log(x)*(x + 3)/(x*x) - 1*3.
(−1)3+2⋅(0+3)log⁡(0)0⋅0\left(-1\right) 3 + \frac{2 \cdot \left(0 + 3\right) \log{\left(0 \right)}}{0 \cdot 0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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(−4log⁡(x)+6(x+3)log⁡(x)x+2−5(x+3)x)x3=0\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=43847.5905884662x_{1} = 43847.5905884662
x2=53841.952602536x_{2} = 53841.952602536
x3=31502.6420619197x_{3} = 31502.6420619197
x4=34886.2640297897x_{4} = 34886.2640297897
x5=52735.3601050521x_{5} = 52735.3601050521
x6=33759.938295641x_{6} = 33759.938295641
x7=2.68147262116116x_{7} = 2.68147262116116
x8=32632.0827716515x_{8} = 32632.0827716515
x9=57156.4807890037x_{9} = 57156.4807890037
x10=30371.5579010177x_{10} = 30371.5579010177
x11=48299.6168667142x_{11} = 48299.6168667142
x12=36011.1126055766x_{12} = 36011.1126055766
x13=40496.6981287632x_{13} = 40496.6981287632
x14=49410.0091923193x_{14} = 49410.0091923193
x15=44962.2264158991x_{15} = 44962.2264158991
x16=50519.4130981396x_{16} = 50519.4130981396
x17=54947.6556019799x_{17} = 54947.6556019799
x18=47188.2084874369x_{18} = 47188.2084874369
x19=37134.5340105345x_{19} = 37134.5340105345
x20=46075.7551136742x_{20} = 46075.7551136742
x21=51627.8549875932x_{21} = 51627.8549875932
x22=41614.8623757666x_{22} = 41614.8623757666
x23=39377.2828114816x_{23} = 39377.2828114816
x24=38256.5757166382x_{24} = 38256.5757166382
x25=42731.8142563409x_{25} = 42731.8142563409
x26=56052.4912533661x_{26} = 56052.4912533661
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(2(−4log⁡(x)+6(x+3)log⁡(x)x+2−5(x+3)x)x3)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty
Let's take the limit
lim⁡x→0+(2(−4log⁡(x)+6(x+3)log⁡(x)x+2−5(x+3)x)x3)=−∞\lim_{x \to 0^+}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty
Let's take the limit
- limits are equal, then skip the corresponding point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[2.68147262116116,∞)\left[2.68147262116116, \infty\right)
Convex at the intervals
(−∞,2.68147262116116]\left(-\infty, 2.68147262116116\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2(x+3)log⁡(x)xx−3)=−3\lim_{x \to -\infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\right) = -3
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−3y = -3
lim⁡x→∞(2(x+3)log⁡(x)xx−3)=−3\lim_{x \to \infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\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 2*log(x)*(x + 3)/(x*x) - 1*3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2(x+3)log⁡(x)xx−3x)=0\lim_{x \to -\infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2(x+3)log⁡(x)xx−3x)=0\lim_{x \to \infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{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:
2(x+3)log⁡(x)xx−3=−3+2⋅(−x+3)log⁡(−x)x2\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = -3 + \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}
- No
2(x+3)log⁡(x)xx−3=3−2⋅(−x+3)log⁡(−x)x2\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 3 - \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = (2lnx/x((x+3)/x))-3
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