The graph y = f(x) = x/(x⁴-16) (x divide by (x to the power of 4 minus 16)) - plot the function graph and draw it. Curve sketching this function. [THERE'S THE ANSWER!] online
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Graphing y = x/(x^4-16)

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The graph:

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Intersection points:

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Piecewise:

The solution

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          x   
f(x) = -------
        4     
       x  - 16
f(x)=xx416f{\left(x \right)} = \frac{x}{x^{4} - 16}
f = x/(x^4 - 16)
The graph of the function
02468-8-6-4-2-10105-5
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = -2
x2=2x_{2} = 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:
xx416=0\frac{x}{x^{4} - 16} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=30448.1852635265x_{1} = -30448.1852635265
x2=13627.3156894593x_{2} = 13627.3156894593
x3=37229.0125605144x_{3} = -37229.0125605144
x4=16170.1475026008x_{4} = 16170.1475026008
x5=9257.90753976151x_{5} = -9257.90753976151
x6=23667.3580679472x_{6} = -23667.3580679472
x7=31427.021528788x_{7} = 31427.021528788
x8=14474.9286237899x_{8} = 14474.9286237899
x9=42314.6338027785x_{9} = -42314.6338027785
x10=33969.831693598x_{10} = 33969.831693598
x11=39055.4525011759x_{11} = 39055.4525011759
x12=42445.8667509803x_{12} = 42445.8667509803
x13=35533.8056321403x_{13} = -35533.8056321403
x14=32143.391993459x_{14} = -32143.391993459
x15=27189.0047552326x_{15} = 27189.0047552326
x16=40619.4266472391x_{16} = -40619.4266472391
x17=20276.9425445156x_{17} = -20276.9425445156
x18=18581.7330944549x_{18} = -18581.7330944549
x19=30579.418167128x_{19} = 30579.418167128
x20=33838.5987785747x_{20} = -33838.5987785747
x21=27057.7718541128x_{21} = -27057.7718541128
x22=21255.7797208243x_{22} = 21255.7797208243
x23=11084.4482824334x_{23} = 11084.4482824334
x24=35665.038553917x_{24} = 35665.038553917
x25=15191.3052003182x_{25} = -15191.3052003182
x26=10953.2106823887x_{26} = -10953.2106823887
x27=40750.6595892069x_{27} = 40750.6595892069
x28=38924.2195657489x_{28} = -38924.2195657489
x29=16886.5212863653x_{29} = -16886.5212863653
x30=32990.9953780887x_{30} = -32990.9953780887
x31=26341.4013776652x_{31} = 26341.4013776652
x32=22103.3836362641x_{32} = 22103.3836362641
x33=10105.5682307074x_{33} = -10105.5682307074
x34=38207.8489855768x_{34} = 38207.8489855768
x35=21972.1506746415x_{35} = -21972.1506746415
x36=25362.5650615x_{36} = -25362.5650615
x37=38076.6160535142x_{37} = -38076.6160535142
x38=28752.97856445x_{38} = -28752.97856445
x39=22819.7544381535x_{39} = -22819.7544381535
x40=39903.0560358041x_{40} = 39903.0560358041
x41=21124.5467305885x_{41} = -21124.5467305885
x42=32274.6249022224x_{42} = 32274.6249022224
x43=12779.699148658x_{43} = 12779.699148658
x44=19560.5711165887x_{44} = 19560.5711165887
x45=22950.9873788779x_{45} = 22950.9873788779
x46=28036.608114504x_{46} = 28036.608114504
x47=13496.0811048387x_{47} = -13496.0811048387
x48=34686.2021962782x_{48} = -34686.2021962782
x49=27905.3752142785x_{49} = -27905.3752142785
x50=41598.2631610587x_{50} = 41598.2631610587
x51=11932.0774603584x_{51} = 11932.0774603584
x52=15322.5389900482x_{52} = 15322.5389900482
x53=0x_{53} = 0
x54=24646.1945150236x_{54} = 24646.1945150236
x55=36512.6420119861x_{55} = 36512.6420119861
x56=36381.40908677x_{56} = -36381.40908677
x57=12648.4639053896x_{57} = -12648.4639053896
x58=9389.15057513904x_{58} = 9389.15057513904
x59=20408.175573558x_{59} = 20408.175573558
x60=23798.5909936427x_{60} = 23798.5909936427
x61=11800.8412650999x_{61} = -11800.8412650999
x62=37360.2454891671x_{62} = 37360.2454891671
x63=29600.5819122437x_{63} = -29600.5819122437
x64=25493.7979695406x_{64} = 25493.7979695406
x65=41467.0302159327x_{65} = -41467.0302159327
x66=17734.1275771308x_{66} = -17734.1275771308
x67=14343.6945028541x_{67} = -14343.6945028541
x68=10236.80795557x_{68} = 10236.80795557
x69=24514.9615998732x_{69} = -24514.9615998732
x70=26210.1684740975x_{70} = -26210.1684740975
x71=19429.3380351854x_{71} = -19429.3380351854
x72=28884.2114649697x_{72} = 28884.2114649697
x73=34817.4351146439x_{73} = 34817.4351146439
x74=39771.8230970718x_{74} = -39771.8230970718
x75=17017.7546618872x_{75} = 17017.7546618872
x76=16038.9139523032x_{76} = -16038.9139523032
x77=33122.2282898894x_{77} = 33122.2282898894
x78=31295.7886227931x_{78} = -31295.7886227931
x79=29731.8148139625x_{79} = 29731.8148139625
x80=18712.966246377x_{80} = 18712.966246377
x81=17865.3608241084x_{81} = 17865.3608241084
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/(x^4 - 16).
016+04\frac{0}{-16 + 0^{4}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
4x4(x416)2+1x416=0- \frac{4 x^{4}}{\left(x^{4} - 16\right)^{2}} + \frac{1}{x^{4} - 16} = 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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
4x3(8x4x4165)(x416)2=0\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=2x_{1} = -2
x2=2x_{2} = 2

limx2(4x3(8x4x4165)(x416)2)=\lim_{x \to -2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty
limx2+(4x3(8x4x4165)(x416)2)=\lim_{x \to -2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty
- the limits are not equal, so
x1=2x_{1} = -2
- is an inflection point
limx2(4x3(8x4x4165)(x416)2)=\lim_{x \to 2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty
limx2+(4x3(8x4x4165)(x416)2)=\lim_{x \to 2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty
- the limits are not equal, so
x2=2x_{2} = 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:
x1=2x_{1} = -2
x2=2x_{2} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
limx(xx416)=0\lim_{x \to -\infty}\left(\frac{x}{x^{4} - 16}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
limx(xx416)=0\lim_{x \to \infty}\left(\frac{x}{x^{4} - 16}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(x^4 - 16), divided by x at x->+oo and x ->-oo
limx1x416=0\lim_{x \to -\infty} \frac{1}{x^{4} - 16} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
limx1x416=0\lim_{x \to \infty} \frac{1}{x^{4} - 16} = 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:
xx416=xx416\frac{x}{x^{4} - 16} = - \frac{x}{x^{4} - 16}
- No
xx416=xx416\frac{x}{x^{4} - 16} = \frac{x}{x^{4} - 16}
- Yes
so, the function
is
odd