Mister Exam
Graphing y = (1-(-1)^x)/(2x+1)
The solution
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x
1 - (-1)
f(x) = ---------
2*x + 1
$$f{\left(x \right)} = \frac{1 - \left(-1\right)^{x}}{2 x + 1}$$
f = (1 - (-1)^x)/(2*x + 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -0.5$$
$$x_{1} = -0.5$$
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:
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -42$$
$$x_{2} = 82$$
$$x_{3} = -66$$
$$x_{4} = 76$$
$$x_{5} = 4$$
$$x_{6} = -30$$
$$x_{7} = 16$$
$$x_{8} = -22$$
$$x_{9} = -76$$
$$x_{10} = 88$$
$$x_{11} = 62$$
$$x_{12} = 8$$
$$x_{13} = 66$$
$$x_{14} = 92$$
$$x_{15} = -62$$
$$x_{16} = 94$$
$$x_{17} = 26$$
$$x_{18} = 46$$
$$x_{19} = 22$$
$$x_{20} = -78$$
$$x_{21} = -48$$
$$x_{22} = 100$$
$$x_{23} = -60$$
$$x_{24} = -90$$
$$x_{25} = -6$$
$$x_{26} = 12$$
$$x_{27} = 32$$
$$x_{28} = 30$$
$$x_{29} = -34$$
$$x_{30} = -20$$
$$x_{31} = -52$$
$$x_{32} = -96$$
$$x_{33} = -64$$
$$x_{34} = 68$$
$$x_{35} = 72$$
$$x_{36} = 50$$
$$x_{37} = -82$$
$$x_{38} = -72$$
$$x_{39} = -24$$
$$x_{40} = 52$$
$$x_{41} = -10$$
$$x_{42} = -8$$
$$x_{43} = 10$$
$$x_{44} = 84$$
$$x_{45} = -28$$
$$x_{46} = -84$$
$$x_{47} = 24$$
$$x_{48} = 64$$
$$x_{49} = 36$$
$$x_{50} = -88$$
$$x_{51} = -2$$
$$x_{52} = 14$$
$$x_{53} = -38$$
$$x_{54} = 60$$
$$x_{55} = 56$$
$$x_{56} = -86$$
$$x_{57} = 40$$
$$x_{58} = -46$$
$$x_{59} = -74$$
$$x_{60} = -26$$
$$x_{61} = -32$$
$$x_{62} = 58$$
$$x_{63} = 70$$
$$x_{64} = -70$$
$$x_{65} = 18$$
$$x_{66} = -68$$
$$x_{67} = -14$$
$$x_{68} = 44$$
$$x_{69} = 6$$
$$x_{70} = -54$$
$$x_{71} = -58$$
$$x_{72} = 90$$
$$x_{73} = 98$$
$$x_{74} = -92$$
$$x_{75} = -94$$
$$x_{76} = 80$$
$$x_{77} = 86$$
$$x_{78} = -44$$
$$x_{79} = -18$$
$$x_{80} = 0$$
$$x_{81} = 48$$
$$x_{82} = 96$$
$$x_{83} = -80$$
$$x_{84} = -16$$
$$x_{85} = 20$$
$$x_{86} = -12$$
$$x_{87} = -56$$
$$x_{88} = 34$$
$$x_{89} = -40$$
$$x_{90} = -50$$
$$x_{91} = 28$$
$$x_{92} = 2$$
$$x_{93} = -36$$
$$x_{94} = 74$$
$$x_{95} = 42$$
$$x_{96} = 54$$
$$x_{97} = -98$$
$$x_{98} = -4$$
$$x_{99} = -100$$
$$x_{100} = 78$$
$$x_{101} = 38$$
so we need to solve the equation:
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -42$$
$$x_{2} = 82$$
$$x_{3} = -66$$
$$x_{4} = 76$$
$$x_{5} = 4$$
$$x_{6} = -30$$
$$x_{7} = 16$$
$$x_{8} = -22$$
$$x_{9} = -76$$
$$x_{10} = 88$$
$$x_{11} = 62$$
$$x_{12} = 8$$
$$x_{13} = 66$$
$$x_{14} = 92$$
$$x_{15} = -62$$
$$x_{16} = 94$$
$$x_{17} = 26$$
$$x_{18} = 46$$
$$x_{19} = 22$$
$$x_{20} = -78$$
$$x_{21} = -48$$
$$x_{22} = 100$$
$$x_{23} = -60$$
$$x_{24} = -90$$
$$x_{25} = -6$$
$$x_{26} = 12$$
$$x_{27} = 32$$
$$x_{28} = 30$$
$$x_{29} = -34$$
$$x_{30} = -20$$
$$x_{31} = -52$$
$$x_{32} = -96$$
$$x_{33} = -64$$
$$x_{34} = 68$$
$$x_{35} = 72$$
$$x_{36} = 50$$
$$x_{37} = -82$$
$$x_{38} = -72$$
$$x_{39} = -24$$
$$x_{40} = 52$$
$$x_{41} = -10$$
$$x_{42} = -8$$
$$x_{43} = 10$$
$$x_{44} = 84$$
$$x_{45} = -28$$
$$x_{46} = -84$$
$$x_{47} = 24$$
$$x_{48} = 64$$
$$x_{49} = 36$$
$$x_{50} = -88$$
$$x_{51} = -2$$
$$x_{52} = 14$$
$$x_{53} = -38$$
$$x_{54} = 60$$
$$x_{55} = 56$$
$$x_{56} = -86$$
$$x_{57} = 40$$
$$x_{58} = -46$$
$$x_{59} = -74$$
$$x_{60} = -26$$
$$x_{61} = -32$$
$$x_{62} = 58$$
$$x_{63} = 70$$
$$x_{64} = -70$$
$$x_{65} = 18$$
$$x_{66} = -68$$
$$x_{67} = -14$$
$$x_{68} = 44$$
$$x_{69} = 6$$
$$x_{70} = -54$$
$$x_{71} = -58$$
$$x_{72} = 90$$
$$x_{73} = 98$$
$$x_{74} = -92$$
$$x_{75} = -94$$
$$x_{76} = 80$$
$$x_{77} = 86$$
$$x_{78} = -44$$
$$x_{79} = -18$$
$$x_{80} = 0$$
$$x_{81} = 48$$
$$x_{82} = 96$$
$$x_{83} = -80$$
$$x_{84} = -16$$
$$x_{85} = 20$$
$$x_{86} = -12$$
$$x_{87} = -56$$
$$x_{88} = 34$$
$$x_{89} = -40$$
$$x_{90} = -50$$
$$x_{91} = 28$$
$$x_{92} = 2$$
$$x_{93} = -36$$
$$x_{94} = 74$$
$$x_{95} = 42$$
$$x_{96} = 54$$
$$x_{97} = -98$$
$$x_{98} = -4$$
$$x_{99} = -100$$
$$x_{100} = 78$$
$$x_{101} = 38$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{\left(-1\right)^{x} i \pi}{2 x + 1} - \frac{2 \left(1 - \left(-1\right)^{x}\right)}{\left(2 x + 1\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{\left(-1\right)^{x} i \pi}{2 x + 1} - \frac{2 \left(1 - \left(-1\right)^{x}\right)}{\left(2 x + 1\right)^{2}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(-1\right)^{x} \pi^{2} + \frac{4 \left(-1\right)^{x} i \pi}{2 x + 1} - \frac{8 \left(\left(-1\right)^{x} - 1\right)}{\left(2 x + 1\right)^{2}}}{2 x + 1} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(-1\right)^{x} \pi^{2} + \frac{4 \left(-1\right)^{x} i \pi}{2 x + 1} - \frac{8 \left(\left(-1\right)^{x} - 1\right)}{\left(2 x + 1\right)^{2}}}{2 x + 1} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -0.5$$
$$x_{1} = -0.5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)$$
Limit on the right could not be calculated
$$\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)$$
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)$$
Limit on the right could not be calculated
$$\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - (-1)^x)/(2*x + 1), divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)$$
Limit on the right could not be calculated
$$\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)$$
Limit on the left could not be calculated
$$\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)$$
Limit on the right could not be calculated
$$\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)$$
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:
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}$$
- No
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = - \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}$$
- No
$$\frac{1 - \left(-1\right)^{x}}{2 x + 1} = - \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}$$
- No
so, the function
not is
neither even, nor odd