Mister Exam

Other calculators

(x+1)sin(x)/(x+1)
Plotting a function graph/ (x+1)sin(x)/(x+1)

Graphing y = (x+1)sin(x)/(x+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       (x + 1)*sin(x)
f(x) = --------------
           x + 1
f(x)=(x+1)sin⁡(x)x+1f{\left(x \right)} = \frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1} 
f = (x + 1)*sin(x)/(x + 1)
The graph of the function
0-80-70-60-50-40-30-20-10102-2
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -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:
(x+1)sin⁡(x)x+1=0\frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=−78.5398163397448x_{1} = -78.5398163397448
x2=−84.8230016469244x_{2} = -84.8230016469244
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−15.707963267949x_{4} = -15.707963267949
x5=21.9911485751286x_{5} = 21.9911485751286
x6=47.1238898038469x_{6} = 47.1238898038469
x7=56.5486677646163x_{7} = 56.5486677646163
x8=−2642.07942166902x_{8} = -2642.07942166902
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=−91.106186954104x_{11} = -91.106186954104
x12=78.5398163397448x_{12} = 78.5398163397448
x13=9.42477796076938x_{13} = 9.42477796076938
x14=15.707963267949x_{14} = 15.707963267949
x15=−87.9645943005142x_{15} = -87.9645943005142
x16=−25.1327412287183x_{16} = -25.1327412287183
x17=37.6991118430775x_{17} = 37.6991118430775
x18=43.9822971502571x_{18} = 43.9822971502571
x19=−81.6814089933346x_{19} = -81.6814089933346
x20=−28.2743338823081x_{20} = -28.2743338823081
x21=−232.477856365645x_{21} = -232.477856365645
x22=−100.530964914873x_{22} = -100.530964914873
x23=18.8495559215388x_{23} = 18.8495559215388
x24=−113.097335529233x_{24} = -113.097335529233
x25=97.3893722612836x_{25} = 97.3893722612836
x26=91.106186954104x_{26} = 91.106186954104
x27=−62.8318530717959x_{27} = -62.8318530717959
x28=−59.6902604182061x_{28} = -59.6902604182061
x29=−267.035375555132x_{29} = -267.035375555132
x30=34.5575191894877x_{30} = 34.5575191894877
x31=−94.2477796076938x_{31} = -94.2477796076938
x32=31.4159265358979x_{32} = 31.4159265358979
x33=94.2477796076938x_{33} = 94.2477796076938
x34=−43.9822971502571x_{34} = -43.9822971502571
x35=−3.14159265358979x_{35} = -3.14159265358979
x36=53.4070751110265x_{36} = 53.4070751110265
x37=81.6814089933346x_{37} = 81.6814089933346
x38=−12.5663706143592x_{38} = -12.5663706143592
x39=62.8318530717959x_{39} = 62.8318530717959
x40=100.530964914873x_{40} = 100.530964914873
x41=−37.6991118430775x_{41} = -37.6991118430775
x42=6.28318530717959x_{42} = 6.28318530717959
x43=−47.1238898038469x_{43} = -47.1238898038469
x44=12.5663706143592x_{44} = 12.5663706143592
x45=28.2743338823081x_{45} = 28.2743338823081
x46=−97.3893722612836x_{46} = -97.3893722612836
x47=75.398223686155x_{47} = 75.398223686155
x48=59.6902604182061x_{48} = 59.6902604182061
x49=87.9645943005142x_{49} = 87.9645943005142
x50=−34.5575191894877x_{50} = -34.5575191894877
x51=−72.2566310325652x_{51} = -72.2566310325652
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=65.9734457253857x_{53} = 65.9734457253857
x54=84.8230016469244x_{54} = 84.8230016469244
x55=−53.4070751110265x_{55} = -53.4070751110265
x56=−69.1150383789755x_{56} = -69.1150383789755
x57=40.8407044966673x_{57} = 40.8407044966673
x58=50.2654824574367x_{58} = 50.2654824574367
x59=25.1327412287183x_{59} = 25.1327412287183
x60=−9.42477796076938x_{60} = -9.42477796076938
x61=−6.28318530717959x_{61} = -6.28318530717959
x62=−65.9734457253857x_{62} = -65.9734457253857
x63=−40.8407044966673x_{63} = -40.8407044966673
x64=69.1150383789755x_{64} = 69.1150383789755
x65=72.2566310325652x_{65} = 72.2566310325652
x66=−75.398223686155x_{66} = -75.398223686155
x67=−56.5486677646163x_{67} = -56.5486677646163
x68=0x_{68} = 0
x69=−18.8495559215388x_{69} = -18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x + 1)*sin(x)/(x + 1).
(0+1)sin⁡(0)0+1\frac{\left(0 + 1\right) \sin{\left(0 \right)}}{0 + 1}
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
cos⁡(x)=0\cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi    
(--, 1)
 2     

 3*pi     
(----, -1)
  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:
Minima of the function at points:
x1=3π2x_{1} = \frac{3 \pi}{2}
Maxima of the function at points:
x1=π2x_{1} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Increasing at intervals
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\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
−sin⁡(x)=0- \sin{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=−1x_{1} = -1

lim⁡x→−1−(−sin⁡(x))=0.841470984807897\lim_{x \to -1^-}\left(- \sin{\left(x \right)}\right) = 0.841470984807897
Let's take the limit
lim⁡x→−1+(−sin⁡(x))=0.841470984807897\lim_{x \to -1^+}\left(- \sin{\left(x \right)}\right) = 0.841470984807897
Let's take the limit
- limits are equal, then skip the corresponding point

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]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Convex at the intervals
[0,π]\left[0, \pi\right]
Vertical asymptotes
Have:
x1=−1x_{1} = -1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x+1)sin⁡(x)x+1)=⟨−1,1⟩\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞((x+1)sin⁡(x)x+1)=⟨−1,1⟩\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x + 1)*sin(x)/(x + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \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:
(x+1)sin⁡(x)x+1=−sin⁡(x)\frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1} = - \sin{\left(x \right)}
- No
(x+1)sin⁡(x)x+1=sin⁡(x)\frac{\left(x + 1\right) \sin{\left(x \right)}}{x + 1} = \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = (x+1)sin(x)/(x+1)
    © Mister Exam – Calculator