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  • How to use it?

  • Graphing y =:
  • x√2-x
  • x^2+4x+2
  • x^2-6x-7
  • x^3-3*x x^3-3*x

  • Identical expressions

  • sqrt(two *cos(x))- one
  • square root of (2 multiply by co sinus of e of (x)) minus 1
  • square root of (two multiply by co sinus of e of (x)) minus one
  • √(2*cos(x))-1
  • sqrt(2cos(x))-1
  • sqrt2cosx-1

  • Similar expressions

  • sqrt(2*cos(x))+1
  • sqrt(2*cosx)-1
Plotting a function graph/ sqrt(2*cos(x))-1

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         __________    
f(x) = \/ 2*cos(x)  - 1
f(x)=2cos⁡(x)−1f{\left(x \right)} = \sqrt{2 \cos{\left(x \right)}} - 1 
f = sqrt(2*cos(x)) - 1
The graph of the function
02468-8-6-4-2-10101-1
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:
2cos⁡(x)−1=0\sqrt{2 \cos{\left(x \right)}} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \pi}{3}
Numerical solution
x1=−99.4837673636768x_{1} = -99.4837673636768
x2=−51.3126800086333x_{2} = -51.3126800086333
x3=−42.9350995990605x_{3} = -42.9350995990605
x4=1.0471975511966x_{4} = 1.0471975511966
x5=11.5191730631626x_{5} = 11.5191730631626
x6=−86.9173967493176x_{6} = -86.9173967493176
x7=49.2182849062401x_{7} = 49.2182849062401
x8=−32.4631240870945x_{8} = -32.4631240870945
x9=−17.8023583703422x_{9} = -17.8023583703422
x10=−38.7463093942741x_{10} = -38.7463093942741
x11=70.162235930172x_{11} = 70.162235930172
x12=−49.2182849062401x_{12} = -49.2182849062401
x13=−26.1799387799149x_{13} = -26.1799387799149
x14=−76.4454212373516x_{14} = -76.4454212373516
x15=36.6519142918809x_{15} = 36.6519142918809
x16=−95.2949771588904x_{16} = -95.2949771588904
x17=5.23598775598299x_{17} = 5.23598775598299
x18=82.7286065445312x_{18} = 82.7286065445312
x19=−13.6135681655558x_{19} = -13.6135681655558
x20=−55.5014702134197x_{20} = -55.5014702134197
x21=95.2949771588904x_{21} = 95.2949771588904
x22=93.2005820564972x_{22} = 93.2005820564972
x23=−5.23598775598299x_{23} = -5.23598775598299
x24=42.9350995990605x_{24} = 42.9350995990605
x25=17.8023583703422x_{25} = 17.8023583703422
x26=63.8790506229925x_{26} = 63.8790506229925
x27=57.5958653158129x_{27} = 57.5958653158129
x28=89.0117918517108x_{28} = 89.0117918517108
x29=19.8967534727354x_{29} = 19.8967534727354
x30=45.0294947014537x_{30} = 45.0294947014537
x31=−57.5958653158129x_{31} = -57.5958653158129
x32=−82.7286065445312x_{32} = -82.7286065445312
x33=−93.2005820564972x_{33} = -93.2005820564972
x34=−19.8967534727354x_{34} = -19.8967534727354
x35=7.33038285837618x_{35} = 7.33038285837618
x36=−89.0117918517108x_{36} = -89.0117918517108
x37=86.9173967493176x_{37} = 86.9173967493176
x38=61.7846555205993x_{38} = 61.7846555205993
x39=−1.0471975511966x_{39} = -1.0471975511966
x40=−68.0678408277789x_{40} = -68.0678408277789
x41=−70.162235930172x_{41} = -70.162235930172
x42=51.3126800086333x_{42} = 51.3126800086333
x43=99.4837673636768x_{43} = 99.4837673636768
x44=−45.0294947014537x_{44} = -45.0294947014537
x45=−36.6519142918809x_{45} = -36.6519142918809
x46=55.5014702134197x_{46} = 55.5014702134197
x47=68.0678408277789x_{47} = 68.0678408277789
x48=74.3510261349584x_{48} = 74.3510261349584
x49=80.634211442138x_{49} = 80.634211442138
x50=76.4454212373516x_{50} = 76.4454212373516
x51=−30.3687289847013x_{51} = -30.3687289847013
x52=24.0855436775217x_{52} = 24.0855436775217
x53=32.4631240870945x_{53} = 32.4631240870945
x54=38.7463093942741x_{54} = 38.7463093942741
x55=−61.7846555205993x_{55} = -61.7846555205993
x56=−9275.02871094827x_{56} = -9275.02871094827
x57=30.3687289847013x_{57} = 30.3687289847013
x58=13.6135681655558x_{58} = 13.6135681655558
x59=−11.5191730631626x_{59} = -11.5191730631626
x60=−24.0855436775217x_{60} = -24.0855436775217
x61=−63.8790506229925x_{61} = -63.8790506229925
x62=26.1799387799149x_{62} = 26.1799387799149
x63=−74.3510261349584x_{63} = -74.3510261349584
x64=−80.634211442138x_{64} = -80.634211442138
x65=−7.33038285837618x_{65} = -7.33038285837618
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(2*cos(x)) - 1.
−1+2cos⁡(0)-1 + \sqrt{2 \cos{\left(0 \right)}}
The result:
f(0)=−1+2f{\left(0 \right)} = -1 + \sqrt{2}
The point:
(0, -1 + sqrt(2))
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
−2cos⁡(x)sin⁡(x)2cos⁡(x)=0- \frac{\sqrt{2} \sqrt{\cos{\left(x \right)}} \sin{\left(x \right)}}{2 \cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
           ___ 
(0, -1 + \/ 2 )

              ___ 
(pi, -1 + I*\/ 2 )


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:
x2=0x_{2} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \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
−2(sin⁡2(x)cos⁡32(x)+2cos⁡(x))4=0- \frac{\sqrt{2} \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}}\right)}{4} = 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→−∞(2cos⁡(x)−1)=2⟨0,1⟩−1\lim_{x \to -\infty}\left(\sqrt{2 \cos{\left(x \right)}} - 1\right) = \sqrt{2} \left\langle 0, 1\right\rangle - 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=2⟨0,1⟩−1y = \sqrt{2} \left\langle 0, 1\right\rangle - 1
lim⁡x→∞(2cos⁡(x)−1)=2⟨0,1⟩−1\lim_{x \to \infty}\left(\sqrt{2 \cos{\left(x \right)}} - 1\right) = \sqrt{2} \left\langle 0, 1\right\rangle - 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=2⟨0,1⟩−1y = \sqrt{2} \left\langle 0, 1\right\rangle - 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(2*cos(x)) - 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2cos⁡(x)−1x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{2 \cos{\left(x \right)}} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2cos⁡(x)−1x)=0\lim_{x \to \infty}\left(\frac{\sqrt{2 \cos{\left(x \right)}} - 1}{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:
2cos⁡(x)−1=2cos⁡(x)−1\sqrt{2 \cos{\left(x \right)}} - 1 = \sqrt{2 \cos{\left(x \right)}} - 1
- Yes
2cos⁡(x)−1=1−2cos⁡(x)\sqrt{2 \cos{\left(x \right)}} - 1 = 1 - \sqrt{2 \cos{\left(x \right)}}
- No
so, the function
is
even
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