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

Graphing y = (2*ln(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 + 3)    
f(x) = ------------ - 3
            x
f(x)=−3+2log⁡(x+3)xf{\left(x \right)} = -3 + \frac{2 \log{\left(x + 3 \right)}}{x} 
f = -3 + (2*log(x + 3))/x
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=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:
−3+2log⁡(x+3)x=0-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−3−2W(−32e92)3x_{1} = -3 - \frac{2 W\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}
x2=−3−2W−1(−32e92)3x_{2} = -3 - \frac{2 W_{-1}\left(- \frac{3}{2 e^{\frac{9}{2}}}\right)}{3}
Numerical solution
x1=−2.98870112007792x_{1} = -2.98870112007792
x2=0.908824451300817x_{2} = 0.908824451300817
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 + 3))/x - 3.
−3+2log⁡(3)0-3 + \frac{2 \log{\left(3 \right)}}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
2x(x+3)−2log⁡(x+3)x2=0\frac{2}{x \left(x + 3\right)} - \frac{2 \log{\left(x + 3 \right)}}{x^{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+3)2−2x(x+3)+2log⁡(x+3)x2)x=0\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x} = 0
Solve this equation
The roots of this equation
x1=54232.7720154785x_{1} = 54232.7720154785
x2=47557.9021762819x_{2} = 47557.9021762819
x3=55341.460585528x_{3} = 55341.460585528
x4=40837.733275076x_{4} = 40837.733275076
x5=46441.2859799058x_{5} = 46441.2859799058
x6=48673.2683225189x_{6} = 48673.2683225189
x7=49787.4275369804x_{7} = 49787.4275369804
x8=50900.4203746904x_{8} = 50900.4203746904
x9=41961.3545187714x_{9} = 41961.3545187714
x10=57555.8805564298x_{10} = 57555.8805564298
x11=35194.474176351x_{11} = 35194.474176351
x12=39712.5339023989x_{12} = 39712.5339023989
x13=32923.8658113025x_{13} = 32923.8658113025
x14=30644.4070335614x_{14} = 30644.4070335614
x15=58661.6683613746x_{15} = 58661.6683613746
x16=−2x_{16} = -2
x17=52012.285038714x_{17} = 52012.285038714
x18=44204.1166985693x_{18} = 44204.1166985693
x19=56449.1537881797x_{19} = 56449.1537881797
x20=53123.0575695818x_{20} = 53123.0575695818
x21=37457.1165432152x_{21} = 37457.1165432152
x22=36326.741841763x_{22} = 36326.741841763
x23=38585.6867389593x_{23} = 38585.6867389593
x24=45323.3738151654x_{24} = 45323.3738151654
x25=43083.4622826575x_{25} = 43083.4622826575
x26=34060.2172304577x_{26} = 34060.2172304577
x27=31785.3046573135x_{27} = 31785.3046573135
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(−1(x+3)2−2x(x+3)+2log⁡(x+3)x2)x)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = -\infty
lim⁡x→0+(2(−1(x+3)2−2x(x+3)+2log⁡(x+3)x2)x)=∞\lim_{x \to 0^+}\left(\frac{2 \left(- \frac{1}{\left(x + 3\right)^{2}} - \frac{2}{x \left(x + 3\right)} + \frac{2 \log{\left(x + 3 \right)}}{x^{2}}\right)}{x}\right) = \infty
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection 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]\left(-\infty, -2\right]
Convex at the intervals
[−2,∞)\left[-2, \infty\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→−∞(−3+2log⁡(x+3)x)=−3\lim_{x \to -\infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\right) = -3
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−3y = -3
lim⁡x→∞(−3+2log⁡(x+3)x)=−3\lim_{x \to \infty}\left(-3 + \frac{2 \log{\left(x + 3 \right)}}{x}\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 + 3))/x - 3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−3+2log⁡(x+3)xx)=0\lim_{x \to -\infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−3+2log⁡(x+3)xx)=0\lim_{x \to \infty}\left(\frac{-3 + \frac{2 \log{\left(x + 3 \right)}}{x}}{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:
−3+2log⁡(x+3)x=−3−2log⁡(3−x)x-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = -3 - \frac{2 \log{\left(3 - x \right)}}{x}
- No
−3+2log⁡(x+3)x=3+2log⁡(3−x)x-3 + \frac{2 \log{\left(x + 3 \right)}}{x} = 3 + \frac{2 \log{\left(3 - x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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