Mister Exam
Graphing y = log(x+1)/log(x-1)
The solution
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log(x + 1)
f(x) = ----------
log(x - 1)
$$f{\left(x \right)} = \frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}$$
f = log(x + 1)/log(x - 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{1} = 2$$
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:
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{\left(x + 1\right) \log{\left(x - 1 \right)}} - \frac{\log{\left(x + 1 \right)}}{\left(x - 1\right) \log{\left(x - 1 \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{\left(x + 1\right) \log{\left(x - 1 \right)}} - \frac{\log{\left(x + 1 \right)}}{\left(x - 1\right) \log{\left(x - 1 \right)}^{2}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 24990.9819022814$$
$$x_{2} = 23012.180065108$$
$$x_{3} = 21696.0720321665$$
$$x_{4} = 11927.1424610773$$
$$x_{5} = 13216.8567271946$$
$$x_{6} = 26974.9097289087$$
$$x_{7} = 27637.2822298$$
$$x_{8} = 18417.9306531122$$
$$x_{9} = 19726.98181305$$
$$x_{10} = 15159.9785186663$$
$$x_{11} = 31621.7160682476$$
$$x_{12} = 17112.093643241$$
$$x_{13} = 30956.4913579122$$
$$x_{14} = 19072.0701365837$$
$$x_{15} = 16460.4647533393$$
$$x_{16} = 30291.7120950935$$
$$x_{17} = 8091.91985555771$$
$$x_{18} = 12571.3885228224$$
$$x_{19} = 10642.6085702957$$
$$x_{20} = 25651.7441696976$$
$$x_{21} = 7458.74655589038$$
$$x_{22} = 10002.4869770251$$
$$x_{23} = 14511.2077796967$$
$$x_{24} = 11284.1895805989$$
$$x_{25} = 8727.02556497286$$
$$x_{26} = 23671.1817540481$$
$$x_{27} = 28300.161960649$$
$$x_{28} = 28963.5353600946$$
$$x_{29} = 29627.3895105595$$
$$x_{30} = 22353.8031957521$$
$$x_{31} = 24330.7885666891$$
$$x_{32} = 13863.4826786086$$
$$x_{33} = 21039.0087570418$$
$$x_{34} = 15809.7458648551$$
$$x_{35} = 20382.6369684156$$
$$x_{36} = 9363.92266180088$$
$$x_{37} = 17764.594137361$$
$$x_{38} = 26313.0587104048$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- 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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 24990.9819022814$$
$$x_{2} = 23012.180065108$$
$$x_{3} = 21696.0720321665$$
$$x_{4} = 11927.1424610773$$
$$x_{5} = 13216.8567271946$$
$$x_{6} = 26974.9097289087$$
$$x_{7} = 27637.2822298$$
$$x_{8} = 18417.9306531122$$
$$x_{9} = 19726.98181305$$
$$x_{10} = 15159.9785186663$$
$$x_{11} = 31621.7160682476$$
$$x_{12} = 17112.093643241$$
$$x_{13} = 30956.4913579122$$
$$x_{14} = 19072.0701365837$$
$$x_{15} = 16460.4647533393$$
$$x_{16} = 30291.7120950935$$
$$x_{17} = 8091.91985555771$$
$$x_{18} = 12571.3885228224$$
$$x_{19} = 10642.6085702957$$
$$x_{20} = 25651.7441696976$$
$$x_{21} = 7458.74655589038$$
$$x_{22} = 10002.4869770251$$
$$x_{23} = 14511.2077796967$$
$$x_{24} = 11284.1895805989$$
$$x_{25} = 8727.02556497286$$
$$x_{26} = 23671.1817540481$$
$$x_{27} = 28300.161960649$$
$$x_{28} = 28963.5353600946$$
$$x_{29} = 29627.3895105595$$
$$x_{30} = 22353.8031957521$$
$$x_{31} = 24330.7885666891$$
$$x_{32} = 13863.4826786086$$
$$x_{33} = 21039.0087570418$$
$$x_{34} = 15809.7458648551$$
$$x_{35} = 20382.6369684156$$
$$x_{36} = 9363.92266180088$$
$$x_{37} = 17764.594137361$$
$$x_{38} = 26313.0587104048$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{\frac{\left(1 + \frac{2}{\log{\left(x - 1 \right)}}\right) \log{\left(x + 1 \right)}}{\left(x - 1\right)^{2} \log{\left(x - 1 \right)}} - \frac{1}{\left(x + 1\right)^{2}} - \frac{2}{\left(x - 1\right) \left(x + 1\right) \log{\left(x - 1 \right)}}}{\log{\left(x - 1 \right)}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{1} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(x + 1)/log(x - 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 1 \right)}}{x \log{\left(x - 1 \right)}}\right) = 0$$
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:
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}$$
- No
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = - \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}$$
- No
$$\frac{\log{\left(x + 1 \right)}}{\log{\left(x - 1 \right)}} = - \frac{\log{\left(1 - x \right)}}{\log{\left(- x - 1 \right)}}$$
- No
so, the function
not is
neither even, nor odd