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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x) - \/ x + 2  + 1
f(x)=(−x+2+cos⁡(x))+1f{\left(x \right)} = \left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1 
f = -sqrt(x + 2) + cos(x) + 1
The graph of the function
2.00.20.40.60.81.01.21.41.61.82-2
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:
(−x+2+cos⁡(x))+1=0\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1 = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=0.822850728246679x_{1} = 0.822850728246679
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x) - sqrt(x + 2) + 1.
(−2+cos⁡(0))+1\left(- \sqrt{2} + \cos{\left(0 \right)}\right) + 1
The result:
f(0)=2−2f{\left(0 \right)} = 2 - \sqrt{2}
The point:
(0, 2 - 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
−sin⁡(x)−12x+2=0- \sin{\left(x \right)} - \frac{1}{2 \sqrt{x + 2}} = 0
Solve this equation
The roots of this equation
x1=43.9084355497083x_{1} = 43.9084355497083
x2=78.5955399542555x_{2} = 78.5955399542555
x3=47.1952370139375x_{3} = 47.1952370139375
x4=119.335113502387x_{4} = 119.335113502387
x5=56.4832396818376x_{5} = 56.4832396818376
x6=34.6402154949931x_{6} = 34.6402154949931
x7=18.7395422089983x_{7} = 18.7395422089983
x8=72.3146642361009x_{8} = 72.3146642361009
x9=12.4343831841409x_{9} = 12.4343831841409
x10=69.0556877155551x_{10} = 69.0556877155551
x11=31.3292099982806x_{11} = 31.3292099982806
x12=37.6195924976803x_{12} = 37.6195924976803
x13=3.35928951718931x_{13} = 3.35928951718931
x14=62.7696855020475x_{14} = 62.7696855020475
x15=87.9118393733129x_{15} = 87.9118393733129
x16=59.7539298783793x_{16} = 59.7539298783793
x17=84.8766710703319x_{17} = 84.8766710703319
x18=50.1962200197561x_{18} = 50.1962200197561
x19=100.481553887293x_{19} = 100.481553887293
x20=94.1967786854416x_{20} = 94.1967786854416
x21=−0.407372935845506x_{21} = -0.407372935845506
x22=53.4742568121558x_{22} = 53.4742568121558
x23=25.0364322230017x_{23} = 25.0364322230017
x24=232.445195585766x_{24} = 232.445195585766
x25=97.4395340030451x_{25} = 97.4395340030451
x26=81.6267056168808x_{26} = 81.6267056168808
x27=40.9171016816084x_{27} = 40.9171016816084
x28=91.1580137466394x_{28} = 91.1580137466394
x29=66.0341016203005x_{29} = 66.0341016203005
x30=15.8266645214722x_{30} = 15.8266645214722
x31=22.0931900652805x_{31} = 22.0931900652805
x32=9.5722930781303x_{32} = 9.5722930781303
x33=6.10666003394995x_{33} = 6.10666003394995
x34=28.3651953409116x_{34} = 28.3651953409116
x35=75.3413386012592x_{35} = 75.3413386012592
The values of the extrema at the points:
(43.90843554970832, -4.77830292793698)

(78.59553995425553, -8.97594971824341)

(47.19523701393752, -7.01138750469409)

(119.33511350238656, -9.01625263569867)

(56.4832396818376, -5.64957318918017)

(34.640215494993065, -6.04969878926072)

(18.73954220899826, -2.56011416068994)

(72.31466423610091, -8.61891189895624)

(12.434383184140883, -1.80795857660106)

(69.05568771555508, -6.4312143171353)

(31.32920999828057, -3.77690311298584)

(37.619592497680316, -4.29756962012607)

(3.3592895171893122, -2.29141140773876)

(62.769685502047516, -6.04989335542023)

(87.91183937331286, -7.48357658753594)

(59.75392987837931, -7.85634065549461)

(84.87667107033185, -9.31932572607707)

(50.19622001975612, -5.22709283452083)

(100.48155388729256, -8.12453781257753)

(94.19677868544157, -7.8092959159218)

(-0.40737293584550616, 0.656171539092775)

(53.47425681215583, -7.4458484010006)

(25.036432223001746, -3.20429106187258)

(232.44519558576573, -13.3121366148542)

(97.4395340030451, -9.97067948807662)

(81.62670561688076, -7.14625969265421)

(40.91710168160843, -6.54819769152871)

(91.15801374663941, -9.65049720746863)

(66.0341016203005, -8.24643970165118)

(15.826664521472244, -4.21512676255686)

(22.09319006528047, -4.90327972493291)

(9.5722930781303, -3.39094668637373)

(6.10666003394995, -0.862759871373676)

(28.36519534091164, -5.50633728955359)

(75.34133860125917, -6.79600997227496)


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:
Minima of the function at points:
x1=78.5955399542555x_{1} = 78.5955399542555
x2=47.1952370139375x_{2} = 47.1952370139375
x3=34.6402154949931x_{3} = 34.6402154949931
x4=72.3146642361009x_{4} = 72.3146642361009
x5=3.35928951718931x_{5} = 3.35928951718931
x6=59.7539298783793x_{6} = 59.7539298783793
x7=84.8766710703319x_{7} = 84.8766710703319
x8=53.4742568121558x_{8} = 53.4742568121558
x9=97.4395340030451x_{9} = 97.4395340030451
x10=40.9171016816084x_{10} = 40.9171016816084
x11=91.1580137466394x_{11} = 91.1580137466394
x12=66.0341016203005x_{12} = 66.0341016203005
x13=15.8266645214722x_{13} = 15.8266645214722
x14=22.0931900652805x_{14} = 22.0931900652805
x15=9.5722930781303x_{15} = 9.5722930781303
x16=28.3651953409116x_{16} = 28.3651953409116
Maxima of the function at points:
x16=43.9084355497083x_{16} = 43.9084355497083
x16=119.335113502387x_{16} = 119.335113502387
x16=56.4832396818376x_{16} = 56.4832396818376
x16=18.7395422089983x_{16} = 18.7395422089983
x16=12.4343831841409x_{16} = 12.4343831841409
x16=69.0556877155551x_{16} = 69.0556877155551
x16=31.3292099982806x_{16} = 31.3292099982806
x16=37.6195924976803x_{16} = 37.6195924976803
x16=62.7696855020475x_{16} = 62.7696855020475
x16=87.9118393733129x_{16} = 87.9118393733129
x16=50.1962200197561x_{16} = 50.1962200197561
x16=100.481553887293x_{16} = 100.481553887293
x16=94.1967786854416x_{16} = 94.1967786854416
x16=−0.407372935845506x_{16} = -0.407372935845506
x16=25.0364322230017x_{16} = 25.0364322230017
x16=232.445195585766x_{16} = 232.445195585766
x16=81.6267056168808x_{16} = 81.6267056168808
x16=6.10666003394995x_{16} = 6.10666003394995
x16=75.3413386012592x_{16} = 75.3413386012592
Decreasing at intervals
[97.4395340030451,∞)\left[97.4395340030451, \infty\right)
Increasing at intervals
(−∞,3.35928951718931]\left(-\infty, 3.35928951718931\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
−cos⁡(x)+14(x+2)32=0- \cos{\left(x \right)} + \frac{1}{4 \left(x + 2\right)^{\frac{3}{2}}} = 0
Solve this equation
The roots of this equation
x1=29.8465212732249x_{1} = 29.8465212732249
x2=89.5351051586156x_{2} = 89.5351051586156
x3=54.9784527026643x_{3} = 54.9784527026643
x4=64.4021873725592x_{4} = 64.4021873725592
x5=17.281712314579x_{5} = 17.281712314579
x6=11.0009074101868x_{6} = 11.0009074101868
x7=58.1189277724023x_{7} = 58.1189277724023
x8=76.9686637605211x_{8} = 76.9686637605211
x9=95.8183175242421x_{9} = 95.8183175242421
x10=83.2518877175584x_{10} = 83.2518877175584
x11=51.8356458841583x_{11} = 51.8356458841583
x12=20.4179969539229x_{12} = 20.4179969539229
x13=4.72671910389813x_{13} = 4.72671910389813
x14=1.53314322401496x_{14} = 1.53314322401496
x15=39.2689651850464x_{15} = 39.2689651850464
x16=36.1293773353517x_{16} = 36.1293773353517
x17=80.1109486658704x_{17} = 80.1109486658704
x18=67.5446731186246x_{18} = 67.5446731186246
x19=61.2615536006668x_{19} = 61.2615536006668
x20=26.701911729892x_{20} = 26.701911729892
x21=42.4123454884565x_{21} = 42.4123454884565
x22=92.6772546569569x_{22} = 92.6772546569569
x23=45.5523310780247x_{23} = 45.5523310780247
x24=32.9855147492632x_{24} = 32.9855147492632
x25=14.1333089972526x_{25} = 14.1333089972526
x26=48.6953787386129x_{26} = 48.6953787386129
x27=73.8278059738963x_{27} = 73.8278059738963
x28=23.5638790964373x_{28} = 23.5638790964373
x29=7.8458895140298x_{29} = 7.8458895140298
x30=70.6854312750525x_{30} = 70.6854312750525
x31=86.3940987926108x_{31} = 86.3940987926108
x32=98.9604150292827x_{32} = 98.9604150292827

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
[95.8183175242421,∞)\left[95.8183175242421, \infty\right)
Convex at the intervals
(−∞,1.53314322401496]\left(-\infty, 1.53314322401496\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((−x+2+cos⁡(x))+1)=⟨0,2⟩−∞i\lim_{x \to -\infty}\left(\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1\right) = \left\langle 0, 2\right\rangle - \infty i
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,2⟩−∞iy = \left\langle 0, 2\right\rangle - \infty i
lim⁡x→∞((−x+2+cos⁡(x))+1)=−∞\lim_{x \to \infty}\left(\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 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 cos(x) - sqrt(x + 2) + 1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((−x+2+cos⁡(x))+1x)=0\lim_{x \to -\infty}\left(\frac{\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((−x+2+cos⁡(x))+1x)=0\lim_{x \to \infty}\left(\frac{\left(- \sqrt{x + 2} + \cos{\left(x \right)}\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:
(−x+2+cos⁡(x))+1=−2−x+cos⁡(x)+1\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1 = - \sqrt{2 - x} + \cos{\left(x \right)} + 1
- No
(−x+2+cos⁡(x))+1=2−x−cos⁡(x)−1\left(- \sqrt{x + 2} + \cos{\left(x \right)}\right) + 1 = \sqrt{2 - x} - \cos{\left(x \right)} - 1
- No
so, the function
not is
neither even, nor odd
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