Mister Exam
Graphing y = sin(x-1)/(x-1)(x-sin(x))
The solution
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sin(x - 1)
f(x) = ----------*(x - sin(x))
x - 1
$$f{\left(x \right)} = \frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right)$$
f = (sin(x - 1)/(x - 1))*(x - sin(x))
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:
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -86.9645943005142$$
$$x_{2} = -71.2566310325652$$
$$x_{3} = -8.42477796076938$$
$$x_{4} = 60.6902604182061$$
$$x_{5} = 44.9822971502571$$
$$x_{6} = 264.893782901543$$
$$x_{7} = -74.398223686155$$
$$x_{8} = 4.14159265358979$$
$$x_{9} = -68.1150383789755$$
$$x_{10} = 7.28318530717959$$
$$x_{11} = 0$$
$$x_{12} = -14.707963267949$$
$$x_{13} = -61.8318530717959$$
$$x_{14} = 32.4159265358979$$
$$x_{15} = -20.9911485751286$$
$$x_{16} = 35.5575191894877$$
$$x_{17} = -77.5398163397448$$
$$x_{18} = 19.8495559215388$$
$$x_{19} = -11.5663706143592$$
$$x_{20} = 98.3893722612836$$
$$x_{21} = 10.4247779607694$$
$$x_{22} = 41.8407044966673$$
$$x_{23} = -42.9822971502571$$
$$x_{24} = 26.1327412287183$$
$$x_{25} = 85.8230016469244$$
$$x_{26} = 48.1238898038469$$
$$x_{27} = 73.2566310325652$$
$$x_{28} = -83.8230016469244$$
$$x_{29} = 29.2743338823081$$
$$x_{30} = -58.6902604182061$$
$$x_{31} = 66.9734457253857$$
$$x_{32} = -30.4159265358979$$
$$x_{33} = -93.2477796076938$$
$$x_{34} = -49.2654824574367$$
$$x_{35} = 16.707963267949$$
$$x_{36} = 13.5663706143592$$
$$x_{37} = 92.106186954104$$
$$x_{38} = 88.9645943005142$$
$$x_{39} = 70.1150383789755$$
$$x_{40} = 22.9911485751286$$
$$x_{41} = 79.5398163397448$$
$$x_{42} = -80.6814089933346$$
$$x_{43} = -90.106186954104$$
$$x_{44} = -1532.09721495182$$
$$x_{45} = -46.1238898038469$$
$$x_{46} = -55.5486677646163$$
$$x_{47} = 38.6991118430775$$
$$x_{48} = -33.5575191894877$$
$$x_{49} = 76.398223686155$$
$$x_{50} = 57.5486677646163$$
$$x_{51} = -2.14159265358979$$
$$x_{52} = -27.2743338823081$$
$$x_{53} = -5.28318530717959$$
$$x_{54} = 63.8318530717959$$
$$x_{55} = -96.3893722612836$$
$$x_{56} = -39.8407044966673$$
$$x_{57} = -64.9734457253857$$
$$x_{58} = -99.5309649148734$$
$$x_{59} = 51.2654824574367$$
$$x_{60} = 54.4070751110265$$
$$x_{61} = 82.6814089933346$$
$$x_{62} = -52.4070751110265$$
$$x_{63} = -17.8495559215388$$
$$x_{64} = 95.2477796076938$$
$$x_{65} = -36.6991118430775$$
$$x_{66} = -24.1327412287183$$
so we need to solve the equation:
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -86.9645943005142$$
$$x_{2} = -71.2566310325652$$
$$x_{3} = -8.42477796076938$$
$$x_{4} = 60.6902604182061$$
$$x_{5} = 44.9822971502571$$
$$x_{6} = 264.893782901543$$
$$x_{7} = -74.398223686155$$
$$x_{8} = 4.14159265358979$$
$$x_{9} = -68.1150383789755$$
$$x_{10} = 7.28318530717959$$
$$x_{11} = 0$$
$$x_{12} = -14.707963267949$$
$$x_{13} = -61.8318530717959$$
$$x_{14} = 32.4159265358979$$
$$x_{15} = -20.9911485751286$$
$$x_{16} = 35.5575191894877$$
$$x_{17} = -77.5398163397448$$
$$x_{18} = 19.8495559215388$$
$$x_{19} = -11.5663706143592$$
$$x_{20} = 98.3893722612836$$
$$x_{21} = 10.4247779607694$$
$$x_{22} = 41.8407044966673$$
$$x_{23} = -42.9822971502571$$
$$x_{24} = 26.1327412287183$$
$$x_{25} = 85.8230016469244$$
$$x_{26} = 48.1238898038469$$
$$x_{27} = 73.2566310325652$$
$$x_{28} = -83.8230016469244$$
$$x_{29} = 29.2743338823081$$
$$x_{30} = -58.6902604182061$$
$$x_{31} = 66.9734457253857$$
$$x_{32} = -30.4159265358979$$
$$x_{33} = -93.2477796076938$$
$$x_{34} = -49.2654824574367$$
$$x_{35} = 16.707963267949$$
$$x_{36} = 13.5663706143592$$
$$x_{37} = 92.106186954104$$
$$x_{38} = 88.9645943005142$$
$$x_{39} = 70.1150383789755$$
$$x_{40} = 22.9911485751286$$
$$x_{41} = 79.5398163397448$$
$$x_{42} = -80.6814089933346$$
$$x_{43} = -90.106186954104$$
$$x_{44} = -1532.09721495182$$
$$x_{45} = -46.1238898038469$$
$$x_{46} = -55.5486677646163$$
$$x_{47} = 38.6991118430775$$
$$x_{48} = -33.5575191894877$$
$$x_{49} = 76.398223686155$$
$$x_{50} = 57.5486677646163$$
$$x_{51} = -2.14159265358979$$
$$x_{52} = -27.2743338823081$$
$$x_{53} = -5.28318530717959$$
$$x_{54} = 63.8318530717959$$
$$x_{55} = -96.3893722612836$$
$$x_{56} = -39.8407044966673$$
$$x_{57} = -64.9734457253857$$
$$x_{58} = -99.5309649148734$$
$$x_{59} = 51.2654824574367$$
$$x_{60} = 54.4070751110265$$
$$x_{61} = 82.6814089933346$$
$$x_{62} = -52.4070751110265$$
$$x_{63} = -17.8495559215388$$
$$x_{64} = 95.2477796076938$$
$$x_{65} = -36.6991118430775$$
$$x_{66} = -24.1327412287183$$
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
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right)\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right)\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right)\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right)\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (sin(x - 1)/(x - 1))*(x - sin(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x - \sin{\left(x \right)}\right) \sin{\left(x - 1 \right)}}{x \left(x - 1\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x - \sin{\left(x \right)}\right) \sin{\left(x - 1 \right)}}{x \left(x - 1\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(x - \sin{\left(x \right)}\right) \sin{\left(x - 1 \right)}}{x \left(x - 1\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(x - \sin{\left(x \right)}\right) \sin{\left(x - 1 \right)}}{x \left(x - 1\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = - \frac{\left(- x + \sin{\left(x \right)}\right) \sin{\left(x + 1 \right)}}{- x - 1}$$
- No
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = \frac{\left(- x + \sin{\left(x \right)}\right) \sin{\left(x + 1 \right)}}{- x - 1}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = - \frac{\left(- x + \sin{\left(x \right)}\right) \sin{\left(x + 1 \right)}}{- x - 1}$$
- No
$$\frac{\sin{\left(x - 1 \right)}}{x - 1} \left(x - \sin{\left(x \right)}\right) = \frac{\left(- x + \sin{\left(x \right)}\right) \sin{\left(x + 1 \right)}}{- x - 1}$$
- No
so, the function
not is
neither even, nor odd