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Plotting a function graph/ ((3x-1)/(2x+11))^(1-3x)

Graphing y = ((3x-1)/(2x+11))^(1-3x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                 1 - 3*x
       /3*x - 1 \       
f(x) = |--------|       
       \2*x + 11/
f(x)=(3x−12x+11)1−3xf{\left(x \right)} = \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} 
f = ((3*x - 1)/(2*x + 11))^(1 - 3*x)
The graph of the function
02468-8-6-4-2-101005e36
The domain of the function
The points at which the function is not precisely defined:
x1=−5.5x_{1} = -5.5
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:
(3x−12x+11)1−3x=0\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=53.7741212127976x_{1} = 53.7741212127976
x2=103.519401791607x_{2} = 103.519401791607
x3=117.501304248323x_{3} = 117.501304248323
x4=93.5380548056062x_{4} = 93.5380548056062
x5=51.8051293241673x_{5} = 51.8051293241673
x6=89.5474867001012x_{6} = 89.5474867001012
x7=97.5298484884927x_{7} = 97.5298484884927
x8=67.637527559058x_{8} = 67.637527559058
x9=77.5861038133537x_{9} = 77.5861038133537
x10=115.503468112155x_{10} = 115.503468112155
x11=47.8820726766363x_{11} = 47.8820726766363
x12=63.666246966531x_{12} = 63.666246966531
x13=36.3585546055207x_{13} = 36.3585546055207
x14=109.510736364186x_{14} = 109.510736364186
x15=73.6038811955179x_{15} = 73.6038811955179
x16=111.508174176918x_{16} = 111.508174176918
x17=40.1360509738206x_{17} = 40.1360509738206
x18=45.9302786975014x_{18} = 45.9302786975014
x19=128.179829125486x_{19} = 128.179829125486
x20=75.5945999804017x_{20} = 75.5945999804017
x21=55.7469763725164x_{21} = 55.7469763725164
x22=107.513452998425x_{22} = 107.513452998425
x23=95.533812303083x_{23} = 95.533812303083
x24=79.3336240071096x_{24} = 79.3336240071096
x25=81.5711319669354x_{25} = 81.5711319669354
x26=65.6511399406226x_{26} = 65.6511399406226
x27=113.505754914447x_{27} = 113.505754914447
x28=101.522663565235x_{28} = 101.522663565235
x29=38.2353877067043x_{29} = 38.2353877067043
x30=99.5261393894545x_{30} = 99.5261393894545
x31=43.9870692663365x_{31} = 43.9870692663365
x32=83.5645162947652x_{32} = 83.5645162947652
x33=59.7019086653428x_{33} = 59.7019086653428
x34=49.8407821921578x_{34} = 49.8407821921578
x35=85.5584023246637x_{35} = 85.5584023246637
x36=87.5527403510542x_{36} = 87.5527403510542
x37=61.6830780131493x_{37} = 61.6830780131493
x38=79.5783061120023x_{38} = 79.5783061120023
x39=71.6140479816439x_{39} = 71.6140479816439
x40=105.51633679067x_{40} = 105.51633679067
x41=69.6252176044538x_{41} = 69.6252176044538
x42=42.0546604871739x_{42} = 42.0546604871739
x43=57.723072470699x_{43} = 57.723072470699
x44=91.5426028689835x_{44} = 91.5426028689835
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((3*x - 1)/(2*x + 11))^(1 - 3*x).
(−1+0⋅30⋅2+11)1−0\left(\frac{-1 + 0 \cdot 3}{0 \cdot 2 + 11}\right)^{1 - 0}
The result:
f(0)=−111f{\left(0 \right)} = - \frac{1}{11}
The point:
(0, -1/11)
Vertical asymptotes
Have:
x1=−5.5x_{1} = -5.5
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3x−12x+11)1−3x=∞\lim_{x \to -\infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(3x−12x+11)1−3x=0\lim_{x \to \infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 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 ((3*x - 1)/(2*x + 11))^(1 - 3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((3x−12x+11)1−3xx)=−∞\lim_{x \to -\infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}}{x}\right) = -\infty
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
lim⁡x→∞((3x−12x+11)1−3xx)=0\lim_{x \to \infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 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:
(3x−12x+11)1−3x=(−3x−111−2x)3x+1\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}
- No
(3x−12x+11)1−3x=−(−3x−111−2x)3x+1\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = - \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}
- No
so, the function
not is
neither even, nor odd
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