Mister Exam
Graphing y = x^2/(x-2)-x
The solution
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2
x
f(x) = ----- - x
x - 2
$$f{\left(x \right)} = \frac{x^{2}}{x - 2} - x$$
f = x^2/(x - 2) - x
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:
$$\frac{x^{2}}{x - 2} - x = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -2.88731568570003 \cdot 10^{36}$$
$$x_{2} = -9.24690192143704 \cdot 10^{32}$$
$$x_{3} = 1.40447783188397 \cdot 10^{31}$$
$$x_{4} = 5.56974830988186 \cdot 10^{32}$$
$$x_{5} = 3.03468741056969 \cdot 10^{32}$$
$$x_{6} = 8.53671991181331 \cdot 10^{31}$$
$$x_{7} = 2.43076702463093 \cdot 10^{30}$$
$$x_{8} = 1.56959495579801 \cdot 10^{29}$$
$$x_{9} = -4.39755093006581 \cdot 10^{27}$$
$$x_{10} = 8.47063306681106 \cdot 10^{28}$$
$$x_{11} = 1.48467256380226 \cdot 10^{35}$$
$$x_{12} = 1.00743967212308 \cdot 10^{33}$$
$$x_{13} = 4.14168304349224 \cdot 10^{32}$$
$$x_{14} = -4.01711249004875 \cdot 10^{27}$$
$$x_{15} = 3.1109042833421 \cdot 10^{33}$$
$$x_{16} = -1.98182490932215 \cdot 10^{31}$$
$$x_{17} = -2.79981192065295 \cdot 10^{27}$$
$$x_{18} = 4.48850821679928 \cdot 10^{35}$$
$$x_{19} = 2.75672669952569 \cdot 10^{34}$$
$$x_{20} = -4.94831242101617 \cdot 10^{35}$$
$$x_{21} = 1.11622037234777 \cdot 10^{35}$$
$$x_{22} = -1.47345006070549 \cdot 10^{32}$$
$$x_{23} = 2.06442057890882 \cdot 10^{29}$$
$$x_{24} = -1.82349077144872 \cdot 10^{27}$$
$$x_{25} = 4.1397735174429 \cdot 10^{33}$$
$$x_{26} = 4.38166555805237 \cdot 10^{35}$$
$$x_{27} = 6.74187268537095 \cdot 10^{35}$$
$$x_{28} = 7.28547293012132 \cdot 10^{30}$$
$$x_{29} = -2.88227670342839 \cdot 10^{28}$$
$$x_{30} = 3.92535271583034 \cdot 10^{35}$$
$$x_{31} = -2.48426015037374 \cdot 10^{34}$$
$$x_{32} = 1.10288694518035 \cdot 10^{30}$$
$$x_{33} = -3.19368233889375 \cdot 10^{34}$$
$$x_{34} = 2.56245175784886 \cdot 10^{27}$$
$$x_{35} = -7.35436291107523 \cdot 10^{28}$$
$$x_{36} = 1.78313009968274 \cdot 10^{33}$$
$$x_{37} = -9.7333957221973 \cdot 10^{29}$$
$$x_{38} = -4.63548580672434 \cdot 10^{30}$$
$$x_{39} = -5.09471679127117 \cdot 10^{32}$$
$$x_{40} = 1.5026028498048 \cdot 10^{29}$$
$$x_{41} = 5.36320395369561 \cdot 10^{33}$$
$$x_{42} = 7.44392515509649 \cdot 10^{32}$$
$$x_{43} = 2.81215235351273 \cdot 10^{28}$$
$$x_{44} = 2.27974489724391 \cdot 10^{34}$$
$$x_{45} = -2.86196916488234 \cdot 10^{33}$$
$$x_{46} = -7.71224354062832 \cdot 10^{31}$$
$$x_{47} = 2.04334633601979 \cdot 10^{27}$$
$$x_{48} = 1.85059222327521 \cdot 10^{27}$$
$$x_{49} = 1.26223871114234 \cdot 10^{28}$$
$$x_{50} = 0$$
$$x_{51} = -8.79394644894212 \cdot 10^{32}$$
$$x_{52} = -4.73776320178797 \cdot 10^{27}$$
$$x_{53} = -4.61446223682556 \cdot 10^{32}$$
$$x_{54} = 5.55277889824185 \cdot 10^{32}$$
$$x_{55} = -2.15514378804013 \cdot 10^{30}$$
$$x_{56} = -5.31199966696363 \cdot 10^{34}$$
$$x_{57} = 3.11212662373426 \cdot 10^{32}$$
$$x_{58} = -3.23609423680662 \cdot 10^{35}$$
$$x_{59} = -3.76184359452026 \cdot 10^{35}$$
$$x_{60} = 4.1250930184767 \cdot 10^{34}$$
$$x_{61} = -2.76327291445603 \cdot 10^{32}$$
$$x_{62} = 8.33853907900608 \cdot 10^{34}$$
$$x_{63} = 1.08550723931963 \cdot 10^{31}$$
$$x_{64} = -1.067697789965 \cdot 10^{35}$$
$$x_{65} = 2.20924148975448 \cdot 10^{31}$$
$$x_{66} = 4.38812347809548 \cdot 10^{31}$$
$$x_{67} = 4.5481674001489 \cdot 10^{27}$$
$$x_{68} = 3.34112702147049 \cdot 10^{28}$$
$$x_{69} = -4.26057530332375 \cdot 10^{29}$$
$$x_{70} = -1.08125467457685 \cdot 10^{28}$$
$$x_{71} = 1.56948550259336 \cdot 10^{34}$$
$$x_{72} = 4.85206755946778 \cdot 10^{29}$$
$$x_{73} = -3.95208610620377 \cdot 10^{31}$$
$$x_{74} = 5.20590333863061 \cdot 10^{30}$$
$$x_{75} = -8.37379996417482 \cdot 10^{33}$$
$$x_{76} = 1.985235738828 \cdot 10^{33}$$
so we need to solve the equation:
$$\frac{x^{2}}{x - 2} - x = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -2.88731568570003 \cdot 10^{36}$$
$$x_{2} = -9.24690192143704 \cdot 10^{32}$$
$$x_{3} = 1.40447783188397 \cdot 10^{31}$$
$$x_{4} = 5.56974830988186 \cdot 10^{32}$$
$$x_{5} = 3.03468741056969 \cdot 10^{32}$$
$$x_{6} = 8.53671991181331 \cdot 10^{31}$$
$$x_{7} = 2.43076702463093 \cdot 10^{30}$$
$$x_{8} = 1.56959495579801 \cdot 10^{29}$$
$$x_{9} = -4.39755093006581 \cdot 10^{27}$$
$$x_{10} = 8.47063306681106 \cdot 10^{28}$$
$$x_{11} = 1.48467256380226 \cdot 10^{35}$$
$$x_{12} = 1.00743967212308 \cdot 10^{33}$$
$$x_{13} = 4.14168304349224 \cdot 10^{32}$$
$$x_{14} = -4.01711249004875 \cdot 10^{27}$$
$$x_{15} = 3.1109042833421 \cdot 10^{33}$$
$$x_{16} = -1.98182490932215 \cdot 10^{31}$$
$$x_{17} = -2.79981192065295 \cdot 10^{27}$$
$$x_{18} = 4.48850821679928 \cdot 10^{35}$$
$$x_{19} = 2.75672669952569 \cdot 10^{34}$$
$$x_{20} = -4.94831242101617 \cdot 10^{35}$$
$$x_{21} = 1.11622037234777 \cdot 10^{35}$$
$$x_{22} = -1.47345006070549 \cdot 10^{32}$$
$$x_{23} = 2.06442057890882 \cdot 10^{29}$$
$$x_{24} = -1.82349077144872 \cdot 10^{27}$$
$$x_{25} = 4.1397735174429 \cdot 10^{33}$$
$$x_{26} = 4.38166555805237 \cdot 10^{35}$$
$$x_{27} = 6.74187268537095 \cdot 10^{35}$$
$$x_{28} = 7.28547293012132 \cdot 10^{30}$$
$$x_{29} = -2.88227670342839 \cdot 10^{28}$$
$$x_{30} = 3.92535271583034 \cdot 10^{35}$$
$$x_{31} = -2.48426015037374 \cdot 10^{34}$$
$$x_{32} = 1.10288694518035 \cdot 10^{30}$$
$$x_{33} = -3.19368233889375 \cdot 10^{34}$$
$$x_{34} = 2.56245175784886 \cdot 10^{27}$$
$$x_{35} = -7.35436291107523 \cdot 10^{28}$$
$$x_{36} = 1.78313009968274 \cdot 10^{33}$$
$$x_{37} = -9.7333957221973 \cdot 10^{29}$$
$$x_{38} = -4.63548580672434 \cdot 10^{30}$$
$$x_{39} = -5.09471679127117 \cdot 10^{32}$$
$$x_{40} = 1.5026028498048 \cdot 10^{29}$$
$$x_{41} = 5.36320395369561 \cdot 10^{33}$$
$$x_{42} = 7.44392515509649 \cdot 10^{32}$$
$$x_{43} = 2.81215235351273 \cdot 10^{28}$$
$$x_{44} = 2.27974489724391 \cdot 10^{34}$$
$$x_{45} = -2.86196916488234 \cdot 10^{33}$$
$$x_{46} = -7.71224354062832 \cdot 10^{31}$$
$$x_{47} = 2.04334633601979 \cdot 10^{27}$$
$$x_{48} = 1.85059222327521 \cdot 10^{27}$$
$$x_{49} = 1.26223871114234 \cdot 10^{28}$$
$$x_{50} = 0$$
$$x_{51} = -8.79394644894212 \cdot 10^{32}$$
$$x_{52} = -4.73776320178797 \cdot 10^{27}$$
$$x_{53} = -4.61446223682556 \cdot 10^{32}$$
$$x_{54} = 5.55277889824185 \cdot 10^{32}$$
$$x_{55} = -2.15514378804013 \cdot 10^{30}$$
$$x_{56} = -5.31199966696363 \cdot 10^{34}$$
$$x_{57} = 3.11212662373426 \cdot 10^{32}$$
$$x_{58} = -3.23609423680662 \cdot 10^{35}$$
$$x_{59} = -3.76184359452026 \cdot 10^{35}$$
$$x_{60} = 4.1250930184767 \cdot 10^{34}$$
$$x_{61} = -2.76327291445603 \cdot 10^{32}$$
$$x_{62} = 8.33853907900608 \cdot 10^{34}$$
$$x_{63} = 1.08550723931963 \cdot 10^{31}$$
$$x_{64} = -1.067697789965 \cdot 10^{35}$$
$$x_{65} = 2.20924148975448 \cdot 10^{31}$$
$$x_{66} = 4.38812347809548 \cdot 10^{31}$$
$$x_{67} = 4.5481674001489 \cdot 10^{27}$$
$$x_{68} = 3.34112702147049 \cdot 10^{28}$$
$$x_{69} = -4.26057530332375 \cdot 10^{29}$$
$$x_{70} = -1.08125467457685 \cdot 10^{28}$$
$$x_{71} = 1.56948550259336 \cdot 10^{34}$$
$$x_{72} = 4.85206755946778 \cdot 10^{29}$$
$$x_{73} = -3.95208610620377 \cdot 10^{31}$$
$$x_{74} = 5.20590333863061 \cdot 10^{30}$$
$$x_{75} = -8.37379996417482 \cdot 10^{33}$$
$$x_{76} = 1.985235738828 \cdot 10^{33}$$
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
$$- \frac{x^{2}}{\left(x - 2\right)^{2}} + \frac{2 x}{x - 2} - 1 = 0$$
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
$$- \frac{x^{2}}{\left(x - 2\right)^{2}} + \frac{2 x}{x - 2} - 1 = 0$$
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
$$\frac{2 \left(\frac{x^{2}}{\left(x - 2\right)^{2}} - \frac{2 x}{x - 2} + 1\right)}{x - 2} = 0$$
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
$$\frac{2 \left(\frac{x^{2}}{\left(x - 2\right)^{2}} - \frac{2 x}{x - 2} + 1\right)}{x - 2} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{x - 2} - x\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2$$
$$\lim_{x \to \infty}\left(\frac{x^{2}}{x - 2} - x\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2$$
$$\lim_{x \to -\infty}\left(\frac{x^{2}}{x - 2} - x\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 2$$
$$\lim_{x \to \infty}\left(\frac{x^{2}}{x - 2} - x\right) = 2$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 2$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^2/(x - 2) - x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{x^{2}}{x - 2} - x}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{x^{2}}{x - 2} - x}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{x^{2}}{x - 2} - x}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{x^{2}}{x - 2} - x}{x}\right) = 0$$
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:
$$\frac{x^{2}}{x - 2} - x = \frac{x^{2}}{- x - 2} + x$$
- No
$$\frac{x^{2}}{x - 2} - x = - \frac{x^{2}}{- x - 2} - x$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x^{2}}{x - 2} - x = \frac{x^{2}}{- x - 2} + x$$
- No
$$\frac{x^{2}}{x - 2} - x = - \frac{x^{2}}{- x - 2} - x$$
- No
so, the function
not is
neither even, nor odd