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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = x - \/  x  - x
f(x)=x−x2−xf{\left(x \right)} = x - \sqrt{x^{2} - x} 
f = x - sqrt(x^2 - x)
The graph of the function
02468-8-6-4-2-1010-2525
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−x2−x=0x - \sqrt{x^{2} - x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=1.62504262805869⋅1028x_{1} = 1.62504262805869 \cdot 10^{28}
x2=1.37955649810789⋅1030x_{2} = 1.37955649810789 \cdot 10^{30}
x3=1.59623125568211⋅1031x_{3} = 1.59623125568211 \cdot 10^{31}
x4=9.06980847762811⋅1031x_{4} = 9.06980847762811 \cdot 10^{31}
x5=1.7164221864142⋅1032x_{5} = 1.7164221864142 \cdot 10^{32}
x6=5.58897734788599⋅1032x_{6} = 5.58897734788599 \cdot 10^{32}
x7=1.09439022896545⋅1031x_{7} = 1.09439022896545 \cdot 10^{31}
x8=2.304591345438⋅1031x_{8} = 2.304591345438 \cdot 10^{31}
x9=7.6321877357881⋅1030x_{9} = 7.6321877357881 \cdot 10^{30}
x10=9.7077991997921⋅1032x_{10} = 9.7077991997921 \cdot 10^{32}
x11=3.77554352063195⋅1028x_{11} = 3.77554352063195 \cdot 10^{28}
x12=2.31825812568162⋅1032x_{12} = 2.31825812568162 \cdot 10^{32}
x13=2.05928325659106⋅1028x_{13} = 2.05928325659106 \cdot 10^{28}
x14=2.76146021831672⋅1027x_{14} = 2.76146021831672 \cdot 10^{27}
x15=1.25369341817483⋅1032x_{15} = 1.25369341817483 \cdot 10^{32}
x16=1.99641097694614⋅1029x_{16} = 1.99641097694614 \cdot 10^{29}
x17=3.32735312594339⋅1029x_{17} = 3.32735312594339 \cdot 10^{29}
x18=2.14529294218284⋅1030x_{18} = 2.14529294218284 \cdot 10^{30}
x19=7.53662826956762⋅1032x_{19} = 7.53662826956762 \cdot 10^{32}
x20=1.09037430017554⋅1028x_{20} = 1.09037430017554 \cdot 10^{28}
x21=8.73637380888125⋅1029x_{21} = 8.73637380888125 \cdot 10^{29}
x22=6.55211009888955⋅1031x_{22} = 6.55211009888955 \cdot 10^{31}
x23=1.31066551931379⋅1027x_{23} = 1.31066551931379 \cdot 10^{27}
x24=7.41847063311719⋅1030x_{24} = 7.41847063311719 \cdot 10^{30}
x25=1.2977405244039⋅1033x_{25} = 1.2977405244039 \cdot 10^{33}
x26=4.97172436641002⋅1030x_{26} = 4.97172436641002 \cdot 10^{30}
x27=1.17301676415335⋅1029x_{27} = 1.17301676415335 \cdot 10^{29}
x28=1.63459608816256⋅1033x_{28} = 1.63459608816256 \cdot 10^{33}
x29=4.18832868866255⋅1032x_{29} = 4.18832868866255 \cdot 10^{32}
x30=5.58765904997679⋅1027x_{30} = 5.58765904997679 \cdot 10^{27}
x31=8.7492894917681⋅1032x_{31} = 8.7492894917681 \cdot 10^{32}
x32=5.44000241662644⋅1029x_{32} = 5.44000241662644 \cdot 10^{29}
x33=0x_{33} = 0
x34=4.66226651004785⋅1031x_{34} = 4.66226651004785 \cdot 10^{31}
x35=6.73762407499375⋅1028x_{35} = 6.73762407499375 \cdot 10^{28}
x36=5.63878719726942⋅1034x_{36} = 5.63878719726942 \cdot 10^{34}
x37=3.29670571199155⋅1031x_{37} = 3.29670571199155 \cdot 10^{31}
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x - sqrt(x^2 - x).
−02−0- \sqrt{0^{2} - 0}
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
−x−12x2−x+1=0- \frac{x - \frac{1}{2}}{\sqrt{x^{2} - x}} + 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
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)24x(x−1)x(x−1)=0\frac{-1 + \frac{\left(2 x - 1\right)^{2}}{4 x \left(x - 1\right)}}{\sqrt{x \left(x - 1\right)}} = 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−x2−x)=−∞\lim_{x \to -\infty}\left(x - \sqrt{x^{2} - x}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x−x2−x)=12\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=12y = \frac{1}{2}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x - sqrt(x^2 - x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x−x2−xx)=2\lim_{x \to -\infty}\left(\frac{x - \sqrt{x^{2} - x}}{x}\right) = 2
Let's take the limit
so,
inclined asymptote equation on the left:
y=2xy = 2 x
lim⁡x→∞(x−x2−xx)=0\lim_{x \to \infty}\left(\frac{x - \sqrt{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:
x−x2−x=−x−x2+xx - \sqrt{x^{2} - x} = - x - \sqrt{x^{2} + x}
- No
x−x2−x=x+x2+xx - \sqrt{x^{2} - x} = x + \sqrt{x^{2} + x}
- No
so, the function
not is
neither even, nor odd
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