Mister Exam
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How to use it?
- Graphing y =:
- x^2-6
-
x^4+8x^2-9
- x^4-2x^2+2
- (x+2)^2
-
Identical expressions
- ln|x+ two +sqrt((x+ two)^ two - one)|
- ln module of x plus 2 plus square root of ((x plus 2) squared minus 1)|
- ln module of x plus two plus square root of ((x plus two) to the power of two minus one)|
- ln|x+2+√((x+2)^2-1)|
- ln|x+2+sqrt((x+2)2-1)|
- ln|x+2+sqrtx+22-1|
- ln|x+2+sqrt((x+2)²-1)|
- ln|x+2+sqrt((x+2) to the power of 2-1)|
- ln|x+2+sqrtx+2^2-1|
-
Similar expressions
- ln|x+2-sqrt((x+2)^2-1)|
- ln|x+2+sqrt((x-2)^2-1)|
- ln|x+2+sqrt((x+2)^2+1)|
- ln|x-2+sqrt((x+2)^2-1)|
Graphing y = ln|x+2+sqrt((x+2)^2-1)|
The solution
You have entered
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f(x) = log\|x + 2 + \/ (x + 2) - 1 |/
$$f{\left(x \right)} = \log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)}$$
f = log(Abs(x + 2 + sqrt((x + 2)^2 - 1)))
The graph of the function
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:
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -2.84680636426264$$
$$x_{2} = -1.918149839521$$
$$x_{3} = -1.20992297199172$$
$$x_{4} = -2.4640608471858$$
$$x_{5} = -2.00675560568004$$
$$x_{6} = -2$$
$$x_{7} = -1.95208538652549$$
$$x_{8} = -1.20898379454618$$
$$x_{9} = -1.86394126032788$$
$$x_{10} = -2.64385163813385$$
$$x_{11} = -2.00124900431934$$
so we need to solve the equation:
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -2.84680636426264$$
$$x_{2} = -1.918149839521$$
$$x_{3} = -1.20992297199172$$
$$x_{4} = -2.4640608471858$$
$$x_{5} = -2.00675560568004$$
$$x_{6} = -2$$
$$x_{7} = -1.95208538652549$$
$$x_{8} = -1.20898379454618$$
$$x_{9} = -1.86394126032788$$
$$x_{10} = -2.64385163813385$$
$$x_{11} = -2.00124900431934$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = \infty$$
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 log(Abs(x + 2 + sqrt((x + 2)^2 - 1))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)}}{x}\right) = 0$$
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:
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = \log{\left(\left|{- x + \sqrt{\left(2 - x\right)^{2} - 1} + 2}\right| \right)}$$
- No
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = - \log{\left(\left|{- x + \sqrt{\left(2 - x\right)^{2} - 1} + 2}\right| \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = \log{\left(\left|{- x + \sqrt{\left(2 - x\right)^{2} - 1} + 2}\right| \right)}$$
- No
$$\log{\left(\left|{\left(x + 2\right) + \sqrt{\left(x + 2\right)^{2} - 1}}\right| \right)} = - \log{\left(\left|{- x + \sqrt{\left(2 - x\right)^{2} - 1} + 2}\right| \right)}$$
- No
so, the function
not is
neither even, nor odd