Mister Exam
Graphing y = (-1)/tan(1+x)
The solution
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-1
f(x) = ----------
tan(1 + x)
$$f{\left(x \right)} = - \frac{1}{\tan{\left(x + 1 \right)}}$$
f = -1/tan(x + 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{1} = -1$$
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:
$$- \frac{1}{\tan{\left(x + 1 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -49.6946861306418$$
$$x_{2} = -74.8274273593601$$
$$x_{3} = -87.3937979737193$$
$$x_{4} = -8.85398163397448$$
$$x_{5} = -71.6858347057703$$
$$x_{6} = 25.7035375555132$$
$$x_{7} = -27.7035375555132$$
$$x_{8} = -84.2522053201295$$
$$x_{9} = 19.4203522483337$$
$$x_{10} = 72.8274273593601$$
$$x_{11} = -21.4203522483337$$
$$x_{12} = -18.2787595947439$$
$$x_{13} = -37.1283155162826$$
$$x_{14} = -24.5619449019235$$
$$x_{15} = 41.4115008234622$$
$$x_{16} = 75.9690200129499$$
$$x_{17} = -59.1194640914112$$
$$x_{18} = 63.4026493985908$$
$$x_{19} = 53.9778714378214$$
$$x_{20} = 60.261056745001$$
$$x_{21} = -5.71238898038469$$
$$x_{22} = 13.1371669411541$$
$$x_{23} = 28.845130209103$$
$$x_{24} = -93.6769832808989$$
$$x_{25} = -33.9867228626928$$
$$x_{26} = -55.9778714378214$$
$$x_{27} = 69.6858347057703$$
$$x_{28} = -90.5353906273091$$
$$x_{29} = 9.99557428756428$$
$$x_{30} = 94.8185759344887$$
$$x_{31} = -65.4026493985908$$
$$x_{32} = 50.8362787842316$$
$$x_{33} = -77.9690200129499$$
$$x_{34} = -99.9601685880785$$
$$x_{35} = 35.1283155162826$$
$$x_{36} = 44.553093477052$$
$$x_{37} = 85.3937979737193$$
$$x_{38} = -68.5442420521806$$
$$x_{39} = 47.6946861306418$$
$$x_{40} = -15.1371669411541$$
$$x_{41} = 57.1194640914112$$
$$x_{42} = 97.9601685880785$$
$$x_{43} = -96.8185759344887$$
$$x_{44} = 0.570796326794897$$
$$x_{45} = 22.5619449019235$$
$$x_{46} = 31.9867228626928$$
$$x_{47} = -46.553093477052$$
$$x_{48} = 88.5353906273091$$
$$x_{49} = -43.4115008234622$$
$$x_{50} = 82.2522053201295$$
$$x_{51} = -52.8362787842316$$
$$x_{52} = 6.85398163397448$$
$$x_{53} = -30.845130209103$$
$$x_{54} = 91.6769832808989$$
$$x_{55} = 79.1106126665397$$
$$x_{56} = 3.71238898038469$$
$$x_{57} = -11.9955742875643$$
$$x_{58} = 38.2699081698724$$
$$x_{59} = 16.2787595947439$$
$$x_{60} = 66.5442420521806$$
$$x_{61} = -81.1106126665397$$
$$x_{62} = -40.2699081698724$$
$$x_{63} = -62.261056745001$$
$$x_{64} = 101.101761241668$$
$$x_{65} = -2.5707963267949$$
so we need to solve the equation:
$$- \frac{1}{\tan{\left(x + 1 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -49.6946861306418$$
$$x_{2} = -74.8274273593601$$
$$x_{3} = -87.3937979737193$$
$$x_{4} = -8.85398163397448$$
$$x_{5} = -71.6858347057703$$
$$x_{6} = 25.7035375555132$$
$$x_{7} = -27.7035375555132$$
$$x_{8} = -84.2522053201295$$
$$x_{9} = 19.4203522483337$$
$$x_{10} = 72.8274273593601$$
$$x_{11} = -21.4203522483337$$
$$x_{12} = -18.2787595947439$$
$$x_{13} = -37.1283155162826$$
$$x_{14} = -24.5619449019235$$
$$x_{15} = 41.4115008234622$$
$$x_{16} = 75.9690200129499$$
$$x_{17} = -59.1194640914112$$
$$x_{18} = 63.4026493985908$$
$$x_{19} = 53.9778714378214$$
$$x_{20} = 60.261056745001$$
$$x_{21} = -5.71238898038469$$
$$x_{22} = 13.1371669411541$$
$$x_{23} = 28.845130209103$$
$$x_{24} = -93.6769832808989$$
$$x_{25} = -33.9867228626928$$
$$x_{26} = -55.9778714378214$$
$$x_{27} = 69.6858347057703$$
$$x_{28} = -90.5353906273091$$
$$x_{29} = 9.99557428756428$$
$$x_{30} = 94.8185759344887$$
$$x_{31} = -65.4026493985908$$
$$x_{32} = 50.8362787842316$$
$$x_{33} = -77.9690200129499$$
$$x_{34} = -99.9601685880785$$
$$x_{35} = 35.1283155162826$$
$$x_{36} = 44.553093477052$$
$$x_{37} = 85.3937979737193$$
$$x_{38} = -68.5442420521806$$
$$x_{39} = 47.6946861306418$$
$$x_{40} = -15.1371669411541$$
$$x_{41} = 57.1194640914112$$
$$x_{42} = 97.9601685880785$$
$$x_{43} = -96.8185759344887$$
$$x_{44} = 0.570796326794897$$
$$x_{45} = 22.5619449019235$$
$$x_{46} = 31.9867228626928$$
$$x_{47} = -46.553093477052$$
$$x_{48} = 88.5353906273091$$
$$x_{49} = -43.4115008234622$$
$$x_{50} = 82.2522053201295$$
$$x_{51} = -52.8362787842316$$
$$x_{52} = 6.85398163397448$$
$$x_{53} = -30.845130209103$$
$$x_{54} = 91.6769832808989$$
$$x_{55} = 79.1106126665397$$
$$x_{56} = 3.71238898038469$$
$$x_{57} = -11.9955742875643$$
$$x_{58} = 38.2699081698724$$
$$x_{59} = 16.2787595947439$$
$$x_{60} = 66.5442420521806$$
$$x_{61} = -81.1106126665397$$
$$x_{62} = -40.2699081698724$$
$$x_{63} = -62.261056745001$$
$$x_{64} = 101.101761241668$$
$$x_{65} = -2.5707963267949$$
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
$$- \frac{- \tan^{2}{\left(x + 1 \right)} - 1}{\tan^{2}{\left(x + 1 \right)}} = 0$$
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
$$- \frac{- \tan^{2}{\left(x + 1 \right)} - 1}{\tan^{2}{\left(x + 1 \right)}} = 0$$
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
$$- \frac{2 \left(\frac{\tan^{2}{\left(x + 1 \right)} + 1}{\tan^{2}{\left(x + 1 \right)}} - 1\right) \left(\tan^{2}{\left(x + 1 \right)} + 1\right)}{\tan{\left(x + 1 \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \left(\frac{\tan^{2}{\left(x + 1 \right)} + 1}{\tan^{2}{\left(x + 1 \right)}} - 1\right) \left(\tan^{2}{\left(x + 1 \right)} + 1\right)}{\tan{\left(x + 1 \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{1} = -1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- \frac{1}{\tan{\left(x + 1 \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(- \frac{1}{\tan{\left(x + 1 \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(- \frac{1}{\tan{\left(x + 1 \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(- \frac{1}{\tan{\left(x + 1 \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -1/tan(1 + x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{1}{x \tan{\left(x + 1 \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{1}{x \tan{\left(x + 1 \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(- \frac{1}{x \tan{\left(x + 1 \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(- \frac{1}{x \tan{\left(x + 1 \right)}}\right)$$
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:
$$- \frac{1}{\tan{\left(x + 1 \right)}} = \frac{1}{\tan{\left(x - 1 \right)}}$$
- No
$$- \frac{1}{\tan{\left(x + 1 \right)}} = - \frac{1}{\tan{\left(x - 1 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \frac{1}{\tan{\left(x + 1 \right)}} = \frac{1}{\tan{\left(x - 1 \right)}}$$
- No
$$- \frac{1}{\tan{\left(x + 1 \right)}} = - \frac{1}{\tan{\left(x - 1 \right)}}$$
- No
so, the function
not is
neither even, nor odd