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-
How to use it?
- Graphing y =:
- x^3-2*x^2+x
- x²-x
- ((x^2+x-6)(x^2-2x-3))/(x^2-9)
- x^3-12x+24
-
Identical expressions
- ((six /(((two two 00x-5 six 25x^2)/(fifty-four - one 70x))+1))^2)/ three ,6
- ((6 divide by (((2200x minus 5625x squared ) divide by (54 minus 170x)) plus 1)) squared ) divide by 3,6
- ((six divide by (((two two 00x minus 5 six 25x squared ) divide by (fifty minus four minus one 70x)) plus 1)) squared ) divide by three ,6
- ((6/(((2200x-5625x2)/(54-170x))+1))2)/3,6
- 6/2200x-5625x2/54-170x+12/3,6
- ((6/(((2200x-5625x²)/(54-170x))+1))²)/3,6
- ((6/(((2200x-5625x to the power of 2)/(54-170x))+1)) to the power of 2)/3,6
- 6/2200x-5625x^2/54-170x+1^2/3,6
- ((6 divide by (((2200x-5625x^2) divide by (54-170x))+1))^2) divide by 3,6
-
Similar expressions
- ((6/(((2200x-5625x^2)/(54+170x))+1))^2)/3,6
- ((6/(((2200x+5625x^2)/(54-170x))+1))^2)/3,6
- ((6/(((2200x-5625x^2)/(54-170x))-1))^2)/3,6
Graphing y = ((6/(((2200x-5625x^2)/(54-170x))+1))^2)/3,6
The solution
You have entered
[src]
2
/ 6 \
|--------------------|
| 2 |
|2200*x - 5625*x |
|---------------- + 1|
\ 54 - 170*x /
f(x) = -----------------------
18/5
$$f{\left(x \right)} = \frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}}$$
f = (6/(1 + (-5625*x^2 + 2200*x)/(54 - 170*x)))^2/(18/5)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -0.0248850408489839$$
$$x_{2} = 0.317647058823529$$
$$x_{3} = 0.385773929737873$$
$$x_{1} = -0.0248850408489839$$
$$x_{2} = 0.317647058823529$$
$$x_{3} = 0.385773929737873$$
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:
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -434208.822559124$$
$$x_{2} = -140986.388074101$$
$$x_{3} = -424097.76733361$$
$$x_{4} = -444319.87633827$$
$$x_{5} = -201653.750411477$$
$$x_{6} = -383653.529692603$$
$$x_{7} = -110652.213183003$$
$$x_{8} = 485891.740234501$$
$$x_{9} = 111779.746971422$$
$$x_{10} = -363431.398883494$$
$$x_{11} = 202781.378174424$$
$$x_{12} = -353320.329884771$$
$$x_{13} = -181431.386146098$$
$$x_{14} = -211764.910006608$$
$$x_{15} = -231987.193860398$$
$$x_{16} = 465669.640745695$$
$$x_{17} = -90429.0270789348$$
$$x_{18} = -191542.576576212$$
$$x_{19} = 475780.691054352$$
$$x_{20} = 324114.758425036$$
$$x_{21} = -373542.465421591$$
$$x_{22} = -454430.928767172$$
$$x_{23} = 0.317647037630407$$
$$x_{24} = -130875.052493621$$
$$x_{25} = 162336.548003835$$
$$x_{26} = 293781.499166832$$
$$x_{27} = -393764.591870289$$
$$x_{28} = 344336.912623802$$
$$x_{29} = -504986.173485526$$
$$x_{30} = 192670.199936394$$
$$x_{31} = 384781.186919614$$
$$x_{32} = -474653.029917949$$
$$x_{33} = 303892.589813626$$
$$x_{34} = -161208.943165154$$
$$x_{35} = 182559.004349482$$
$$x_{36} = -322987.105828175$$
$$x_{37} = -494875.126628933$$
$$x_{38} = 496002.788355432$$
$$x_{39} = -343209.258209209$$
$$x_{40} = 91556.4943274001$$
$$x_{41} = 243225.961261785$$
$$x_{42} = -282542.755962014$$
$$x_{43} = 334225.837161525$$
$$x_{44} = 172447.788326988$$
$$x_{45} = -151097.682409676$$
$$x_{46} = -484764.078793831$$
$$x_{47} = -312876.024542778$$
$$x_{48} = -262320.551575427$$
$$x_{49} = -464541.979933618$$
$$x_{50} = -292653.85003391$$
$$x_{51} = 121891.220787613$$
$$x_{52} = 152225.278474707$$
$$x_{53} = 445447.536434102$$
$$x_{54} = 506113.835480876$$
$$x_{55} = -171320.17621699$$
$$x_{56} = 101668.181792869$$
$$x_{57} = 223003.692126826$$
$$x_{58} = 374670.122030155$$
$$x_{59} = -120763.665478808$$
$$x_{60} = 273559.302830349$$
$$x_{61} = 283670.403685716$$
$$x_{62} = 354447.985093633$$
$$x_{63} = 455558.589233062$$
$$x_{64} = -272431.656675334$$
$$x_{65} = 435336.482258781$$
$$x_{66} = 233114.831561332$$
$$x_{67} = 415114.369374837$$
$$x_{68} = -100540.676299919$$
$$x_{69} = 132002.624549964$$
$$x_{70} = 212892.541558438$$
$$x_{71} = 364559.05482136$$
$$x_{72} = -333098.183614499$$
$$x_{73} = 405003.310438756$$
$$x_{74} = 263448.195977291$$
$$x_{75} = -403875.652111096$$
$$x_{76} = 253337.082402794$$
$$x_{77} = -242098.321046413$$
$$x_{78} = 314003.676096045$$
$$x_{79} = 142113.973418865$$
$$x_{80} = 425225.426608457$$
$$x_{81} = -252209.439968645$$
$$x_{82} = -413986.71055626$$
$$x_{83} = -302764.939410044$$
$$x_{84} = -221876.057290954$$
$$x_{85} = 394892.249668786$$
so we need to solve the equation:
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -434208.822559124$$
$$x_{2} = -140986.388074101$$
$$x_{3} = -424097.76733361$$
$$x_{4} = -444319.87633827$$
$$x_{5} = -201653.750411477$$
$$x_{6} = -383653.529692603$$
$$x_{7} = -110652.213183003$$
$$x_{8} = 485891.740234501$$
$$x_{9} = 111779.746971422$$
$$x_{10} = -363431.398883494$$
$$x_{11} = 202781.378174424$$
$$x_{12} = -353320.329884771$$
$$x_{13} = -181431.386146098$$
$$x_{14} = -211764.910006608$$
$$x_{15} = -231987.193860398$$
$$x_{16} = 465669.640745695$$
$$x_{17} = -90429.0270789348$$
$$x_{18} = -191542.576576212$$
$$x_{19} = 475780.691054352$$
$$x_{20} = 324114.758425036$$
$$x_{21} = -373542.465421591$$
$$x_{22} = -454430.928767172$$
$$x_{23} = 0.317647037630407$$
$$x_{24} = -130875.052493621$$
$$x_{25} = 162336.548003835$$
$$x_{26} = 293781.499166832$$
$$x_{27} = -393764.591870289$$
$$x_{28} = 344336.912623802$$
$$x_{29} = -504986.173485526$$
$$x_{30} = 192670.199936394$$
$$x_{31} = 384781.186919614$$
$$x_{32} = -474653.029917949$$
$$x_{33} = 303892.589813626$$
$$x_{34} = -161208.943165154$$
$$x_{35} = 182559.004349482$$
$$x_{36} = -322987.105828175$$
$$x_{37} = -494875.126628933$$
$$x_{38} = 496002.788355432$$
$$x_{39} = -343209.258209209$$
$$x_{40} = 91556.4943274001$$
$$x_{41} = 243225.961261785$$
$$x_{42} = -282542.755962014$$
$$x_{43} = 334225.837161525$$
$$x_{44} = 172447.788326988$$
$$x_{45} = -151097.682409676$$
$$x_{46} = -484764.078793831$$
$$x_{47} = -312876.024542778$$
$$x_{48} = -262320.551575427$$
$$x_{49} = -464541.979933618$$
$$x_{50} = -292653.85003391$$
$$x_{51} = 121891.220787613$$
$$x_{52} = 152225.278474707$$
$$x_{53} = 445447.536434102$$
$$x_{54} = 506113.835480876$$
$$x_{55} = -171320.17621699$$
$$x_{56} = 101668.181792869$$
$$x_{57} = 223003.692126826$$
$$x_{58} = 374670.122030155$$
$$x_{59} = -120763.665478808$$
$$x_{60} = 273559.302830349$$
$$x_{61} = 283670.403685716$$
$$x_{62} = 354447.985093633$$
$$x_{63} = 455558.589233062$$
$$x_{64} = -272431.656675334$$
$$x_{65} = 435336.482258781$$
$$x_{66} = 233114.831561332$$
$$x_{67} = 415114.369374837$$
$$x_{68} = -100540.676299919$$
$$x_{69} = 132002.624549964$$
$$x_{70} = 212892.541558438$$
$$x_{71} = 364559.05482136$$
$$x_{72} = -333098.183614499$$
$$x_{73} = 405003.310438756$$
$$x_{74} = 263448.195977291$$
$$x_{75} = -403875.652111096$$
$$x_{76} = 253337.082402794$$
$$x_{77} = -242098.321046413$$
$$x_{78} = 314003.676096045$$
$$x_{79} = 142113.973418865$$
$$x_{80} = 425225.426608457$$
$$x_{81} = -252209.439968645$$
$$x_{82} = -413986.71055626$$
$$x_{83} = -302764.939410044$$
$$x_{84} = -221876.057290954$$
$$x_{85} = 394892.249668786$$
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
$$\frac{120 \cdot \left(\frac{1}{6} + \frac{- 5625 x^{2} + 2200 x}{6 \cdot \left(- 170 x + 54\right)}\right) \left(- \frac{- 11250 x + 2200}{- 170 x + 54} - \frac{170 \left(- 5625 x^{2} + 2200 x\right)}{\left(- 170 x + 54\right)^{2}}\right)}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{4}} = 0$$
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
$$\frac{120 \cdot \left(\frac{1}{6} + \frac{- 5625 x^{2} + 2200 x}{6 \cdot \left(- 170 x + 54\right)}\right) \left(- \frac{- 11250 x + 2200}{- 170 x + 54} - \frac{170 \left(- 5625 x^{2} + 2200 x\right)}{\left(- 170 x + 54\right)^{2}}\right)}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{4}} = 0$$
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
$$- \frac{10000 \cdot \left(2 \cdot \left(\frac{25 x \left(225 x - 88\right)}{85 x - 27} + 2\right) \left(\frac{1445 x \left(225 x - 88\right)}{\left(85 x - 27\right)^{2}} + 45 - \frac{34 \cdot \left(225 x - 44\right)}{85 x - 27}\right) - \frac{15 \left(- 450 x + \frac{85 x \left(225 x - 88\right)}{85 x - 27} + 88\right)^{2}}{85 x - 27}\right)}{\left(85 x - 27\right) \left(\frac{25 x \left(225 x - 88\right)}{85 x - 27} + 2\right)^{4}} = 0$$
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
$$- \frac{10000 \cdot \left(2 \cdot \left(\frac{25 x \left(225 x - 88\right)}{85 x - 27} + 2\right) \left(\frac{1445 x \left(225 x - 88\right)}{\left(85 x - 27\right)^{2}} + 45 - \frac{34 \cdot \left(225 x - 44\right)}{85 x - 27}\right) - \frac{15 \left(- 450 x + \frac{85 x \left(225 x - 88\right)}{85 x - 27} + 88\right)^{2}}{85 x - 27}\right)}{\left(85 x - 27\right) \left(\frac{25 x \left(225 x - 88\right)}{85 x - 27} + 2\right)^{4}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -0.0248850408489839$$
$$x_{2} = 0.317647058823529$$
$$x_{3} = 0.385773929737873$$
$$x_{1} = -0.0248850408489839$$
$$x_{2} = 0.317647058823529$$
$$x_{3} = 0.385773929737873$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}}\right) = 0$$
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 (6/((2200*x - 5625*x^2)/(54 - 170*x) + 1))^2/(18/5), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{5 \cdot \frac{36}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{2}}}{18 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{5 \cdot \frac{36}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{2}}}{18 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{5 \cdot \frac{36}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{2}}}{18 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{5 \cdot \frac{36}{\left(1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}\right)^{2}}}{18 x}\right) = 0$$
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:
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = \frac{10}{\left(1 + \frac{\left(-1\right) 5625 x^{2} - 2200 x}{170 x + 54}\right)^{2}}$$
- No
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = - \frac{10}{\left(1 + \frac{\left(-1\right) 5625 x^{2} - 2200 x}{170 x + 54}\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = \frac{10}{\left(1 + \frac{\left(-1\right) 5625 x^{2} - 2200 x}{170 x + 54}\right)^{2}}$$
- No
$$\frac{\left(\frac{6}{1 + \frac{- 5625 x^{2} + 2200 x}{- 170 x + 54}}\right)^{2}}{\frac{18}{5}} = - \frac{10}{\left(1 + \frac{\left(-1\right) 5625 x^{2} - 2200 x}{170 x + 54}\right)^{2}}$$
- No
so, the function
not is
neither even, nor odd
The graph