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Plotting a function graph/ sqrt(2x-1)-x

Graphing y = sqrt(2x-1)-x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = \/ 2*x - 1  - x
f(x)=−x+2x−1f{\left(x \right)} = - x + \sqrt{2 x - 1} 
f = -x + sqrt(2*x - 1)
The graph of the function
02468-8-6-4-2-10105-10
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:
−x+2x−1=0- x + \sqrt{2 x - 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1.00000097440894x_{1} = 1.00000097440894
x2=1.00000117987389x_{2} = 1.00000117987389
x3=1.00000097876361x_{3} = 1.00000097876361
x4=1.00000105919658x_{4} = 1.00000105919658
x5=1.00000116462766x_{5} = 1.00000116462766
x6=1.00000090690937x_{6} = 1.00000090690937
x7=1.00000055333913x_{7} = 1.00000055333913
x8=1.00000083172526x_{8} = 1.00000083172526
x9=1.00000109482874x_{9} = 1.00000109482874
x10=1.00000070881331x_{10} = 1.00000070881331
x11=1.00000099128414x_{11} = 1.00000099128414
x12=1.00000096546994x_{12} = 1.00000096546994
x13=1.00000112960777x_{13} = 1.00000112960777
x14=1.0000011468244x_{14} = 1.0000011468244
x15=1.00000116634587x_{15} = 1.00000116634587
x16=1.00000101630877x_{16} = 1.00000101630877
x17=1.00000059655034x_{17} = 1.00000059655034
x18=1.00000100658205x_{18} = 1.00000100658205
x19=1.00000101432834x_{19} = 1.00000101432834
x20=1.00000110754817x_{20} = 1.00000110754817
x21=1.00000096250645x_{21} = 1.00000096250645
x22=0.99999989747815x_{22} = 0.99999989747815
x23=1.00000093650649x_{23} = 1.00000093650649
x24=1.00000106682114x_{24} = 1.00000106682114
x25=1.00000114164492x_{25} = 1.00000114164492
x26=1.0000002724443x_{26} = 1.0000002724443
x27=1.00000109519652x_{27} = 1.00000109519652
x28=1.00000062155742x_{28} = 1.00000062155742
x29=1.00000107149836x_{29} = 1.00000107149836
x30=1.00000086200244x_{30} = 1.00000086200244
x31=1.00000112315959x_{31} = 1.00000112315959
x32=1.00000090221369x_{32} = 1.00000090221369
x33=1.00000078317785x_{33} = 1.00000078317785
x34=1.00000065962627x_{34} = 1.00000065962627
x35=1.00000112528943x_{35} = 1.00000112528943
x36=1.00000110128469x_{36} = 1.00000110128469
x37=1.00000110254157x_{37} = 1.00000110254157
x38=1.00000110964016x_{38} = 1.00000110964016
x39=1.00000108816829x_{39} = 1.00000108816829
x40=1.00000089042622x_{40} = 1.00000089042622
x41=1.0000008725758x_{41} = 1.0000008725758
x42=1.00000121247297x_{42} = 1.00000121247297
x43=1.00000099638865x_{43} = 1.00000099638865
x44=1.00000080799155x_{44} = 1.00000080799155
x45=1.00000105128639x_{45} = 1.00000105128639
x46=1.00000078134283x_{46} = 1.00000078134283
x47=1.0000009499013x_{47} = 1.0000009499013
x48=1.00000113633387x_{48} = 1.00000113633387
x49=1.0000007490939x_{49} = 1.0000007490939
x50=1.00000045442477x_{50} = 1.00000045442477
x51=1.00000117095353x_{51} = 1.00000117095353
x52=1.00000104306865x_{52} = 1.00000104306865
x53=1.00000102498357x_{53} = 1.00000102498357
x54=1.00000118509788x_{54} = 1.00000118509788
x55=1.00000113088447x_{55} = 1.00000113088447
x56=1.00000113586449x_{56} = 1.00000113586449
x57=1.00000119923415x_{57} = 1.00000119923415
x58=1.00000106297248x_{58} = 1.00000106297248
x59=1.00000111650806x_{59} = 1.00000111650806
x60=1.00000103451879x_{60} = 1.00000103451879
x61=1.00000114194065x_{61} = 1.00000114194065
x62=1.00000050877525x_{62} = 1.00000050877525
x63=1.00000105409068x_{63} = 1.00000105409068
x64=1.00000071568972x_{64} = 1.00000071568972
x65=1.00000107969536x_{65} = 1.00000107969536
x66=1.00000091984005x_{66} = 1.00000091984005
x67=1.00000116163405x_{67} = 1.00000116163405
x68=1.00000083873372x_{68} = 1.00000083873372
x69=1.00000120815282x_{69} = 1.00000120815282
x70=1.00000115187858x_{70} = 1.00000115187858
x71=1.00000114784621x_{71} = 1.00000114784621
x72=1.00000111954085x_{72} = 1.00000111954085
x73=1.00000067361783x_{73} = 1.00000067361783
x74=1.0000011946276x_{74} = 1.0000011946276
x75=1.00000117511288x_{75} = 1.00000117511288
x76=1.00000118016495x_{76} = 1.00000118016495
x77=1.00000115918158x_{77} = 1.00000115918158
x78=1.00000095130306x_{78} = 1.00000095130306
x79=1.00000098568217x_{79} = 1.00000098568217
x80=1.00000115359037x_{80} = 1.00000115359037
x81=1.00000103513321x_{81} = 1.00000103513321
x82=1.00000120374118x_{82} = 1.00000120374118
x83=1.00000117546148x_{83} = 1.00000117546148
x84=1.00000093614226x_{84} = 1.00000093614226
x85=1.00000088303301x_{85} = 1.00000088303301
x86=1.00000111363019x_{86} = 1.00000111363019
x87=1.00000075097317x_{87} = 1.00000075097317
x88=1.00000118419467x_{88} = 1.00000118419467
x89=1.00000107417975x_{89} = 1.00000107417975
x90=1.00000118991708x_{90} = 1.00000118991708
x91=1.00000081270274x_{91} = 1.00000081270274
x92=1.00000104482245x_{92} = 1.00000104482245
x93=1.00000116993586x_{93} = 1.00000116993586
x94=1.00000100311539x_{94} = 1.00000100311539
x95=1.00000108758756x_{95} = 1.00000108758756
x96=1.00000092221789x_{96} = 1.00000092221789
x97=1.00000108129013x_{97} = 1.00000108129013
x98=1.00000085311401x_{98} = 1.00000085311401
x99=1.00000102560922x_{99} = 1.00000102560922
x100=1.00000036436214x_{100} = 1.00000036436214
x101=1.00000115681331x_{101} = 1.00000115681331
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(2*x - 1) - x.
−0+−1+0⋅2- 0 + \sqrt{-1 + 0 \cdot 2}
The result:
f(0)=if{\left(0 \right)} = i
The point:
(0, i)
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
−1+12x−1=0-1 + \frac{1}{\sqrt{2 x - 1}} = 0
Solve this equation
The roots of this equation
x1=1x_{1} = 1
The values of the extrema at the points:
(1, 0)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x1=1x_{1} = 1
Decreasing at intervals
(−∞,1]\left(-\infty, 1\right]
Increasing at intervals
[1,∞)\left[1, \infty\right)
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
−1(2x−1)32=0- \frac{1}{\left(2 x - 1\right)^{\frac{3}{2}}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(−x+2x−1)=∞\lim_{x \to -\infty}\left(- x + \sqrt{2 x - 1}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(−x+2x−1)=−∞\lim_{x \to \infty}\left(- x + \sqrt{2 x - 1}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(2*x - 1) - x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−x+2x−1x)=−1\lim_{x \to -\infty}\left(\frac{- x + \sqrt{2 x - 1}}{x}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the left:
y=−xy = - x
lim⁡x→∞(−x+2x−1x)=−1\lim_{x \to \infty}\left(\frac{- x + \sqrt{2 x - 1}}{x}\right) = -1
Let's take the limit
so,
inclined asymptote equation on the right:
y=−xy = - x
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:
−x+2x−1=x+−2x−1- x + \sqrt{2 x - 1} = x + \sqrt{- 2 x - 1}
- No
−x+2x−1=−x−−2x−1- x + \sqrt{2 x - 1} = - x - \sqrt{- 2 x - 1}
- No
so, the function
not is
neither even, nor odd
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