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  • Graphing y =:
  • x^2+3x+3
  • x^2-3x-4
  • x^2-2x-2
  • (x^2+3)/(x-1) (x^2+3)/(x-1)

  • Identical expressions

  • atan(five ^x+ one)^ three
  • arc tangent of gent of (5 to the power of x plus 1) cubed
  • arc tangent of gent of (five to the power of x plus one) to the power of three
  • atan(5x+1)3
  • atan5x+13
  • atan(5^x+1)³
  • atan(5 to the power of x+1) to the power of 3
  • atan5^x+1^3

  • Similar expressions

  • atan(5^x-1)^3
  • arctan(5^x+1)^3
Plotting a function graph/ atan(5^x+1)^3

Graphing y = atan(5^x+1)^3

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           3/ x    \
f(x) = atan \5  + 1/
f(x)=atan⁡3(5x+1)f{\left(x \right)} = \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} 
f = atan(5^x + 1)^3
The graph of the function
02468-8-6-4-2-101005
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:
atan⁡3(5x+1)=0\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(5^x + 1)^3.
atan⁡3(50+1)\operatorname{atan}^{3}{\left(5^{0} + 1 \right)}
The result:
f(0)=atan⁡3(2)f{\left(0 \right)} = \operatorname{atan}^{3}{\left(2 \right)}
The point:
(0, atan(2)^3)
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
3⋅5xlog⁡(5)atan⁡2(5x+1)(5x+1)2+1=0\frac{3 \cdot 5^{x} \log{\left(5 \right)} \operatorname{atan}^{2}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 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
3⋅5x(−2⋅5x(5x+1)atan⁡(5x+1)(5x+1)2+1+2⋅5x(5x+1)2+1+atan⁡(5x+1))log⁡(5)2atan⁡(5x+1)(5x+1)2+1=0\frac{3 \cdot 5^{x} \left(- \frac{2 \cdot 5^{x} \left(5^{x} + 1\right) \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} + \frac{2 \cdot 5^{x}}{\left(5^{x} + 1\right)^{2} + 1} + \operatorname{atan}{\left(5^{x} + 1 \right)}\right) \log{\left(5 \right)}^{2} \operatorname{atan}{\left(5^{x} + 1 \right)}}{\left(5^{x} + 1\right)^{2} + 1} = 0
Solve this equation
The roots of this equation
x1=95.2242768391714x_{1} = 95.2242768391714
x2=85.2242768391714x_{2} = 85.2242768391714
x3=21.2242894893908x_{3} = 21.2242894893908
x4=−44.9148275920459x_{4} = -44.9148275920459
x5=−60.9148275920459x_{5} = -60.9148275920459
x6=−36.9148275920459x_{6} = -36.9148275920459
x7=−66.9148275920459x_{7} = -66.9148275920459
x8=−28.9148275920415x_{8} = -28.9148275920415
x9=93.2242768391714x_{9} = 93.2242768391714
x10=−110.914827592046x_{10} = -110.914827592046
x11=−46.9148275920459x_{11} = -46.9148275920459
x12=43.2242768391714x_{12} = 43.2242768391714
x13=−84.9148275920459x_{13} = -84.9148275920459
x14=−90.9148275920459x_{14} = -90.9148275920459
x15=−102.914827592046x_{15} = -102.914827592046
x16=69.2242768391714x_{16} = 69.2242768391714
x17=−112.914827592046x_{17} = -112.914827592046
x18=31.2242768391727x_{18} = 31.2242768391727
x19=−20.9148258621738x_{19} = -20.9148258621738
x20=−82.9148275920459x_{20} = -82.9148275920459
x21=−56.9148275920459x_{21} = -56.9148275920459
x22=39.2242768391714x_{22} = 39.2242768391714
x23=87.2242768391714x_{23} = 87.2242768391714
x24=−70.9148275920459x_{24} = -70.9148275920459
x25=−76.9148275920459x_{25} = -76.9148275920459
x26=53.2242768391714x_{26} = 53.2242768391714
x27=71.2242768391714x_{27} = 71.2242768391714
x28=23.2242773451776x_{28} = 23.2242773451776
x29=103.224276839171x_{29} = 103.224276839171
x30=−104.914827592046x_{30} = -104.914827592046
x31=81.2242768391714x_{31} = 81.2242768391714
x32=−68.9148275920459x_{32} = -68.9148275920459
x33=19.2245931342562x_{33} = 19.2245931342562
x34=0.542176246771954x_{34} = 0.542176246771954
x35=79.2242768391714x_{35} = 79.2242768391714
x36=−38.9148275920459x_{36} = -38.9148275920459
x37=27.224276839981x_{37} = 27.224276839981
x38=55.2242768391714x_{38} = 55.2242768391714
x39=−26.9148275919352x_{39} = -26.9148275919352
x40=−64.9148275920459x_{40} = -64.9148275920459
x41=−50.9148275920459x_{41} = -50.9148275920459
x42=−42.9148275920459x_{42} = -42.9148275920459
x43=41.2242768391714x_{43} = 41.2242768391714
x44=67.2242768391714x_{44} = 67.2242768391714
x45=63.2242768391714x_{45} = 63.2242768391714
x46=−54.9148275920459x_{46} = -54.9148275920459
x47=45.2242768391714x_{47} = 45.2242768391714
x48=75.2242768391714x_{48} = 75.2242768391714
x49=−92.9148275920459x_{49} = -92.9148275920459
x50=−100.914827592046x_{50} = -100.914827592046
x51=97.2242768391714x_{51} = 97.2242768391714
x52=107.224276839171x_{52} = 107.224276839171
x53=101.224276839171x_{53} = 101.224276839171
x54=−52.9148275920459x_{54} = -52.9148275920459
x55=−32.9148275920459x_{55} = -32.9148275920459
x56=−74.9148275920459x_{56} = -74.9148275920459
x57=105.224276839171x_{57} = 105.224276839171
x58=83.2242768391714x_{58} = 83.2242768391714
x59=−80.9148275920459x_{59} = -80.9148275920459
x60=91.2242768391714x_{60} = 91.2242768391714
x61=51.2242768391714x_{61} = 51.2242768391714
x62=−96.9148275920459x_{62} = -96.9148275920459
x63=77.2242768391714x_{63} = 77.2242768391714
x64=−40.9148275920459x_{64} = -40.9148275920459
x65=−72.9148275920459x_{65} = -72.9148275920459
x66=−24.914827589278x_{66} = -24.914827589278
x67=61.2242768391714x_{67} = 61.2242768391714
x68=−78.9148275920459x_{68} = -78.9148275920459
x69=37.2242768391714x_{69} = 37.2242768391714
x70=113.224276839171x_{70} = 113.224276839171
x71=−94.9148275920459x_{71} = -94.9148275920459
x72=109.224276839171x_{72} = 109.224276839171
x73=47.2242768391714x_{73} = 47.2242768391714
x74=111.224276839171x_{74} = 111.224276839171
x75=57.2242768391714x_{75} = 57.2242768391714
x76=−108.914827592046x_{76} = -108.914827592046
x77=−22.9148275228492x_{77} = -22.9148275228492
x78=25.2242768594116x_{78} = 25.2242768594116
x79=89.2242768391714x_{79} = 89.2242768391714
x80=29.2242768392038x_{80} = 29.2242768392038
x81=59.2242768391714x_{81} = 59.2242768391714
x82=35.2242768391714x_{82} = 35.2242768391714
x83=99.2242768391714x_{83} = 99.2242768391714
x84=−86.9148275920459x_{84} = -86.9148275920459
x85=−106.914827592046x_{85} = -106.914827592046
x86=49.2242768391714x_{86} = 49.2242768391714
x87=−98.9148275920459x_{87} = -98.9148275920459
x88=−62.9148275920459x_{88} = -62.9148275920459
x89=33.2242768391715x_{89} = 33.2242768391715
x90=65.2242768391714x_{90} = 65.2242768391714
x91=−88.9148275920459x_{91} = -88.9148275920459
x92=−48.9148275920459x_{92} = -48.9148275920459
x93=−58.9148275920459x_{93} = -58.9148275920459
x94=73.2242768391714x_{94} = 73.2242768391714
x95=−18.914784374042x_{95} = -18.914784374042
x96=−30.9148275920457x_{96} = -30.9148275920457
x97=−34.9148275920459x_{97} = -34.9148275920459

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
(−∞,0.542176246771954]\left(-\infty, 0.542176246771954\right]
Convex at the intervals
[0.542176246771954,∞)\left[0.542176246771954, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞atan⁡3(5x+1)=π364\lim_{x \to -\infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{64}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=π364y = \frac{\pi^{3}}{64}
lim⁡x→∞atan⁡3(5x+1)=π38\lim_{x \to \infty} \operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \frac{\pi^{3}}{8}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=π38y = \frac{\pi^{3}}{8}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(5^x + 1)^3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡3(5x+1)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡3(5x+1)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}^{3}{\left(5^{x} + 1 \right)}}{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:
atan⁡3(5x+1)=atan⁡3(1+5−x)\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}
- No
atan⁡3(5x+1)=−atan⁡3(1+5−x)\operatorname{atan}^{3}{\left(5^{x} + 1 \right)} = - \operatorname{atan}^{3}{\left(1 + 5^{- x} \right)}
- No
so, the function
not is
neither even, nor odd
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