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Plotting a function graph/ log(6*x)/(x+6)

Graphing y = log(6*x)/(x+6)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(6*x)
f(x) = --------
        x + 6
f(x)=log⁡(6x)x+6f{\left(x \right)} = \frac{\log{\left(6 x \right)}}{x + 6} 
f = log(6*x)/(x + 6)
The graph of the function
02468-8-6-4-2-10100.5-0.5
The domain of the function
The points at which the function is not precisely defined:
x1=−6x_{1} = -6
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:
log⁡(6x)x+6=0\frac{\log{\left(6 x \right)}}{x + 6} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=16x_{1} = \frac{1}{6}
Numerical solution
x1=0.166666666666667x_{1} = 0.166666666666667
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(6*x)/(x + 6).
log⁡(0⋅6)6\frac{\log{\left(0 \cdot 6 \right)}}{6}
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
−log⁡(6x)(x+6)2+1x(x+6)=0- \frac{\log{\left(6 x \right)}}{\left(x + 6\right)^{2}} + \frac{1}{x \left(x + 6\right)} = 0
Solve this equation
The roots of this equation
x1=e1+W(36e)6x_{1} = \frac{e^{1 + W\left(\frac{36}{e}\right)}}{6}
The values of the extrema at the points:
       /    -1\                     
  1 + W\36*e  /         /    -1\    
 e                 1 + W\36*e  /    
(--------------, ------------------)
       6                   /    -1\ 
                      1 + W\36*e  / 
                     e              
                 6 + -------------- 
                           6


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=e1+W(36e)6x_{1} = \frac{e^{1 + W\left(\frac{36}{e}\right)}}{6}
Decreasing at intervals
(−∞,e1+W(36e)6]\left(-\infty, \frac{e^{1 + W\left(\frac{36}{e}\right)}}{6}\right]
Increasing at intervals
[e1+W(36e)6,∞)\left[\frac{e^{1 + W\left(\frac{36}{e}\right)}}{6}, \infty\right)
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
2log⁡(6x)(x+6)2−2x(x+6)−1x2x+6=0\frac{\frac{2 \log{\left(6 x \right)}}{\left(x + 6\right)^{2}} - \frac{2}{x \left(x + 6\right)} - \frac{1}{x^{2}}}{x + 6} = 0
Solve this equation
The roots of this equation
x1=42514.0197126849x_{1} = 42514.0197126849
x2=44595.665429855x_{2} = 44595.665429855
x3=54989.4041088202x_{3} = 54989.4041088202
x4=50835.1171488098x_{4} = 50835.1171488098
x5=6.69656678575869x_{5} = 6.69656678575869
x6=46676.4702816404x_{6} = 46676.4702816404
x7=40431.7067631467x_{7} = 40431.7067631467
x8=27949.5664707624x_{8} = 27949.5664707624
x9=26914.6814181565x_{9} = 26914.6814181565
x10=38348.9767867252x_{10} = 38348.9767867252
x11=49795.8499616273x_{11} = 49795.8499616273
x12=30024.098021514x_{12} = 30024.098021514
x13=47716.5183302058x_{13} = 47716.5183302058
x14=37307.5625704563x_{14} = 37307.5625704563
x15=25882.0101643839x_{15} = 25882.0101643839
x16=36266.1905436718x_{16} = 36266.1905436718
x17=53951.2568344188x_{17} = 53951.2568344188
x18=32102.7899925164x_{18} = 32102.7899925164
x19=35224.9316224061x_{19} = 35224.9316224061
x20=43554.9383137055x_{20} = 43554.9383137055
x21=51874.1102610085x_{21} = 51874.1102610085
x22=39390.3744480501x_{22} = 39390.3744480501
x23=33143.1152657172x_{23} = 33143.1152657172
x24=34183.871705079x_{24} = 34183.871705079
x25=41472.9331284706x_{25} = 41472.9331284706
x26=52912.8246738068x_{26} = 52912.8246738068
x27=31063.0528403066x_{27} = 31063.0528403066
x28=48756.3145754334x_{28} = 48756.3145754334
x29=58102.1214499038x_{29} = 58102.1214499038
x30=57064.8372940295x_{30} = 57064.8372940295
x31=45636.1815004787x_{31} = 45636.1815004787
x32=56027.2646514781x_{32} = 56027.2646514781
x33=28986.1676948273x_{33} = 28986.1676948273
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=−6x_{1} = -6

lim⁡x→−6−(2log⁡(6x)(x+6)2−2x(x+6)−1x2x+6)=−∞sign⁡(7.16703787691222+2iπ)\lim_{x \to -6^-}\left(\frac{\frac{2 \log{\left(6 x \right)}}{\left(x + 6\right)^{2}} - \frac{2}{x \left(x + 6\right)} - \frac{1}{x^{2}}}{x + 6}\right) = - \infty \operatorname{sign}{\left(7.16703787691222 + 2 i \pi \right)}
lim⁡x→−6+(2log⁡(6x)(x+6)2−2x(x+6)−1x2x+6)=∞sign⁡(7.16703787691222+2iπ)\lim_{x \to -6^+}\left(\frac{\frac{2 \log{\left(6 x \right)}}{\left(x + 6\right)^{2}} - \frac{2}{x \left(x + 6\right)} - \frac{1}{x^{2}}}{x + 6}\right) = \infty \operatorname{sign}{\left(7.16703787691222 + 2 i \pi \right)}
- the limits are not equal, so
x1=−6x_{1} = -6
- 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
[6.69656678575869,∞)\left[6.69656678575869, \infty\right)
Convex at the intervals
(−∞,6.69656678575869]\left(-\infty, 6.69656678575869\right]
Vertical asymptotes
Have:
x1=−6x_{1} = -6
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(6x)x+6)=0\lim_{x \to -\infty}\left(\frac{\log{\left(6 x \right)}}{x + 6}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(log⁡(6x)x+6)=0\lim_{x \to \infty}\left(\frac{\log{\left(6 x \right)}}{x + 6}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(6*x)/(x + 6), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(6x)x(x+6))=0\lim_{x \to -\infty}\left(\frac{\log{\left(6 x \right)}}{x \left(x + 6\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(6x)x(x+6))=0\lim_{x \to \infty}\left(\frac{\log{\left(6 x \right)}}{x \left(x + 6\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:
log⁡(6x)x+6=log⁡(−6x)6−x\frac{\log{\left(6 x \right)}}{x + 6} = \frac{\log{\left(- 6 x \right)}}{6 - x}
- No
log⁡(6x)x+6=−log⁡(−6x)6−x\frac{\log{\left(6 x \right)}}{x + 6} = - \frac{\log{\left(- 6 x \right)}}{6 - x}
- No
so, the function
not is
neither even, nor odd
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