Mister Exam
Graphing y = ((3x-1)/(2x+11))^(1-3x)
The solution
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1 - 3*x
/3*x - 1 \
f(x) = |--------|
\2*x + 11/
$$f{\left(x \right)} = \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}$$
f = ((3*x - 1)/(2*x + 11))^(1 - 3*x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -5.5$$
$$x_{1} = -5.5$$
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:
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 53.7741212127976$$
$$x_{2} = 103.519401791607$$
$$x_{3} = 117.501304248323$$
$$x_{4} = 93.5380548056062$$
$$x_{5} = 51.8051293241673$$
$$x_{6} = 89.5474867001012$$
$$x_{7} = 97.5298484884927$$
$$x_{8} = 67.637527559058$$
$$x_{9} = 77.5861038133537$$
$$x_{10} = 115.503468112155$$
$$x_{11} = 47.8820726766363$$
$$x_{12} = 63.666246966531$$
$$x_{13} = 36.3585546055207$$
$$x_{14} = 109.510736364186$$
$$x_{15} = 73.6038811955179$$
$$x_{16} = 111.508174176918$$
$$x_{17} = 40.1360509738206$$
$$x_{18} = 45.9302786975014$$
$$x_{19} = 128.179829125486$$
$$x_{20} = 75.5945999804017$$
$$x_{21} = 55.7469763725164$$
$$x_{22} = 107.513452998425$$
$$x_{23} = 95.533812303083$$
$$x_{24} = 79.3336240071096$$
$$x_{25} = 81.5711319669354$$
$$x_{26} = 65.6511399406226$$
$$x_{27} = 113.505754914447$$
$$x_{28} = 101.522663565235$$
$$x_{29} = 38.2353877067043$$
$$x_{30} = 99.5261393894545$$
$$x_{31} = 43.9870692663365$$
$$x_{32} = 83.5645162947652$$
$$x_{33} = 59.7019086653428$$
$$x_{34} = 49.8407821921578$$
$$x_{35} = 85.5584023246637$$
$$x_{36} = 87.5527403510542$$
$$x_{37} = 61.6830780131493$$
$$x_{38} = 79.5783061120023$$
$$x_{39} = 71.6140479816439$$
$$x_{40} = 105.51633679067$$
$$x_{41} = 69.6252176044538$$
$$x_{42} = 42.0546604871739$$
$$x_{43} = 57.723072470699$$
$$x_{44} = 91.5426028689835$$
so we need to solve the equation:
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 53.7741212127976$$
$$x_{2} = 103.519401791607$$
$$x_{3} = 117.501304248323$$
$$x_{4} = 93.5380548056062$$
$$x_{5} = 51.8051293241673$$
$$x_{6} = 89.5474867001012$$
$$x_{7} = 97.5298484884927$$
$$x_{8} = 67.637527559058$$
$$x_{9} = 77.5861038133537$$
$$x_{10} = 115.503468112155$$
$$x_{11} = 47.8820726766363$$
$$x_{12} = 63.666246966531$$
$$x_{13} = 36.3585546055207$$
$$x_{14} = 109.510736364186$$
$$x_{15} = 73.6038811955179$$
$$x_{16} = 111.508174176918$$
$$x_{17} = 40.1360509738206$$
$$x_{18} = 45.9302786975014$$
$$x_{19} = 128.179829125486$$
$$x_{20} = 75.5945999804017$$
$$x_{21} = 55.7469763725164$$
$$x_{22} = 107.513452998425$$
$$x_{23} = 95.533812303083$$
$$x_{24} = 79.3336240071096$$
$$x_{25} = 81.5711319669354$$
$$x_{26} = 65.6511399406226$$
$$x_{27} = 113.505754914447$$
$$x_{28} = 101.522663565235$$
$$x_{29} = 38.2353877067043$$
$$x_{30} = 99.5261393894545$$
$$x_{31} = 43.9870692663365$$
$$x_{32} = 83.5645162947652$$
$$x_{33} = 59.7019086653428$$
$$x_{34} = 49.8407821921578$$
$$x_{35} = 85.5584023246637$$
$$x_{36} = 87.5527403510542$$
$$x_{37} = 61.6830780131493$$
$$x_{38} = 79.5783061120023$$
$$x_{39} = 71.6140479816439$$
$$x_{40} = 105.51633679067$$
$$x_{41} = 69.6252176044538$$
$$x_{42} = 42.0546604871739$$
$$x_{43} = 57.723072470699$$
$$x_{44} = 91.5426028689835$$
Vertical asymptotes
Have:
$$x_{1} = -5.5$$
$$x_{1} = -5.5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x}}{x}\right) = 0$$
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:
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}$$
- No
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = - \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}$$
- No
$$\left(\frac{3 x - 1}{2 x + 11}\right)^{1 - 3 x} = - \left(\frac{- 3 x - 1}{11 - 2 x}\right)^{3 x + 1}$$
- No
so, the function
not is
neither even, nor odd