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Plotting a function graph/ (x-1)*cosx/x-1

Graphing y = (x-1)*cosx/x-1

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       (x - 1)*cos(x)    
f(x) = -------------- - 1
             x
f(x)=−1+(x−1)cos⁡(x)xf{\left(x \right)} = -1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} 
f = -1 + ((x - 1)*cos(x))/x
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
−1+(x−1)cos⁡(x)x=0-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−3908.11864147039x_{1} = -3908.11864147039
x2=−81.5255757034701x_{2} = -81.5255757034701
x3=−37.4706068235463x_{3} = -37.4706068235463
x4=−87.8143902428644x_{4} = -87.8143902428644
x5=−50.4629398821162x_{5} = -50.4629398821162
x6=−81.8369485363712x_{6} = -81.8369485363712
x7=−6.79528546545564x_{7} = -6.79528546545564
x8=−88.1145447112556x_{8} = -88.1145447112556
x9=−31.6640081143412x_{9} = -31.6640081143412
x10=−94.1026353004828x_{10} = -94.1026353004828
x11=−63.0088484909748x_{11} = -63.0088484909748
x12=−56.7350583803468x_{12} = -56.7350583803468
x13=−50.0672590311525x_{13} = -50.0672590311525
x14=−75.5600271487453x_{14} = -75.5600271487453
x15=−43.7705436902596x_{15} = -43.7705436902596
x16=−1.03674878997284x_{16} = -1.03674878997284
x17=−69.2839281169777x_{17} = -69.2839281169777
x18=−12.174233004382x_{18} = -12.174233004382
x19=−150.681555978239x_{19} = -150.681555978239
x20=−24.8537031333593x_{20} = -24.8537031333593
x21=−18.528148876584x_{21} = -18.528148876584
x22=−37.9262696941923x_{22} = -37.9262696941923
x23=−100.390400893298x_{23} = -100.390400893298
x24=−19.1657971772164x_{24} = -19.1657971772164
x25=−56.3616697214293x_{25} = -56.3616697214293
x26=−62.6543642083992x_{26} = -62.6543642083992
x27=−68.9457398660007x_{27} = -68.9457398660007
x28=−44.193054743776x_{28} = -44.193054743776
x29=−25.4088121091366x_{29} = -25.4088121091366
x30=−31.165921585635x_{30} = -31.165921585635
x31=−94.3927026839426x_{31} = -94.3927026839426
x32=−12.9473476761254x_{32} = -12.9473476761254
x33=−75.2360760574743x_{33} = -75.2360760574743
x34=−5830.81448414035x_{34} = -5830.81448414035
x35=−5.73110630283233x_{35} = -5.73110630283233
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((x - 1)*cos(x))/x - 1.
−1+(−1)cos⁡(0)0-1 + \frac{\left(-1\right) \cos{\left(0 \right)}}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−(x−1)sin⁡(x)+cos⁡(x)x−(x−1)cos⁡(x)x2=0\frac{- \left(x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{x} - \frac{\left(x - 1\right) \cos{\left(x \right)}}{x^{2}} = 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
−(x−1)cos⁡(x)−2sin⁡(x)+(x−1)sin⁡(x)x+(x−1)sin⁡(x)−cos⁡(x)x−cos⁡(x)x+2(x−1)cos⁡(x)x2x=0\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x} = 0
Solve this equation
The roots of this equation
x1=29.8474528425252x_{1} = 29.8474528425252
x2=−14.1278030885708x_{2} = -14.1278030885708
x3=4.81830169068692x_{3} = 4.81830169068692
x4=32.9886180185199x_{4} = 32.9886180185199
x5=−80.11030486752x_{5} = -80.11030486752
x6=−865.508773397218x_{6} = -865.508773397218
x7=−58.1188820004811x_{7} = -58.1188820004811
x8=76.9693620488425x_{8} = 76.9693620488425
x9=54.9785453606757x_{9} = 54.9785453606757
x10=64.4031391874027x_{10} = 64.4031391874027
x11=−76.9686867419196x_{11} = -76.9686867419196
x12=73.8277993317264x_{12} = 73.8277993317264
x13=61.2615984927811x_{13} = 61.2615984927811
x14=−32.9849386326173x_{14} = -32.9849386326173
x15=−230.907022689846x_{15} = -230.907022689846
x16=−98.9599664047991x_{16} = -98.9599664047991
x17=67.5446870137729x_{17} = 67.5446870137729
x18=−92.6767529095687x_{18} = -92.6767529095687
x19=83.252497386865x_{19} = 83.252497386865
x20=−29.8429571923873x_{20} = -29.8429571923873
x21=−17.2724203975121x_{21} = -17.2724203975121
x22=17.2858609706125x_{22} = 17.2858609706125
x23=−39.2686433436629x_{23} = -39.2686433436629
x24=14.1479101581108x_{24} = 14.1479101581108
x25=−70.6854400018162x_{25} = -70.6854400018162
x26=23.565705260759x_{26} = 23.565705260759
x27=80.1109282396762x_{27} = 80.1109282396762
x28=39.2712388323218x_{28} = 39.2712388323218
x29=−42.4104144616263x_{29} = -42.4104144616263
x30=98.9603748966698x_{30} = 98.9603748966698
x31=−64.4021745660819x_{31} = -64.4021745660819
x32=86.394069065896x_{32} = 86.394069065896
x33=2.1151044019306x_{33} = 2.1151044019306
x34=−7.82491949822255x_{34} = -7.82491949822255
x35=58.1200665270198x_{35} = 58.1200665270198
x36=70.6862407240204x_{36} = 70.6862407240204
x37=−67.543810056952x_{37} = -67.543810056952
x38=48.6955472336803x_{38} = 48.6955472336803
x39=−23.5584875548962x_{39} = -23.5584875548962
x40=95.8187960668258x_{40} = 95.8187960668258
x41=−86.3935330805343x_{41} = -86.3935330805343
x42=45.5540788615421x_{42} = 45.5540788615421
x43=−10.9803508603381x_{43} = -10.9803508603381
x44=51.8370377167161x_{44} = 51.8370377167161
x45=−45.5521503065062x_{45} = -45.5521503065062
x46=11.0136769372905x_{46} = 11.0136769372905
x47=−36.1268243364932x_{47} = -36.1268243364932
x48=−95.8183603465263x_{48} = -95.8183603465263
x49=20.4253916500379x_{49} = 20.4253916500379
x50=−48.6938595973699x_{50} = -48.6938595973699
x51=−89.5351438981654x_{51} = -89.5351438981654
x52=−61.2605323722968x_{52} = -61.2605323722968
x53=92.6772186743975x_{53} = 92.6772186743975
x54=−51.8355485164882x_{54} = -51.8355485164882
x55=42.4126394726376x_{55} = 42.4126394726376
x56=−4.63467435481449x_{56} = -4.63467435481449
x57=1198.517598738x_{57} = 1198.517598738
x58=−73.8270653183329x_{58} = -73.8270653183329
x59=−26.7008332529908x_{59} = -26.7008332529908
x60=−20.4157768960442x_{60} = -20.4157768960442
x61=36.1298911934334x_{61} = 36.1298911934334
x62=26.7064504389974x_{62} = 26.7064504389974
x63=−54.9772215461932x_{63} = -54.9772215461932
x64=89.5356429258046x_{64} = 89.5356429258046
x65=7.89057914345282x_{65} = 7.89057914345282
x66=−83.2519201806158x_{66} = -83.2519201806158
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=0x_{1} = 0

lim⁡x→0−(−(x−1)cos⁡(x)−2sin⁡(x)+(x−1)sin⁡(x)x+(x−1)sin⁡(x)−cos⁡(x)x−cos⁡(x)x+2(x−1)cos⁡(x)x2x)=∞\lim_{x \to 0^-}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = \infty
lim⁡x→0+(−(x−1)cos⁡(x)−2sin⁡(x)+(x−1)sin⁡(x)x+(x−1)sin⁡(x)−cos⁡(x)x−cos⁡(x)x+2(x−1)cos⁡(x)x2x)=−∞\lim_{x \to 0^+}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty
- the limits are not equal, so
x1=0x_{1} = 0
- 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:
Concave at the intervals
[95.8187960668258,∞)\left[95.8187960668258, \infty\right)
Convex at the intervals
(−∞,−865.508773397218]\left(-\infty, -865.508773397218\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−1+(x−1)cos⁡(x)x)=⟨−2,0⟩\lim_{x \to -\infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,0⟩y = \left\langle -2, 0\right\rangle
lim⁡x→∞(−1+(x−1)cos⁡(x)x)=⟨−2,0⟩\lim_{x \to \infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,0⟩y = \left\langle -2, 0\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((x - 1)*cos(x))/x - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−1+(x−1)cos⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−1+(x−1)cos⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{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:
−1+(x−1)cos⁡(x)x=−1−(−x−1)cos⁡(x)x-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = -1 - \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}
- No
−1+(x−1)cos⁡(x)x=1+(−x−1)cos⁡(x)x-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 1 + \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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