Mister Exam
Other calculators
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-
How to use it?
- Graphing y =:
- x^3+x^2-x+1
- x^3-12x+24
- (x-3)/(x-1)
- (x+3)/(x+2)
-
Identical expressions
- abs(two -(abs(log3abs(x/ two))))
- abs(2 minus (abs( logarithm of 3abs(x divide by 2))))
- abs(two minus (abs( logarithm of 3abs(x divide by two))))
- abs2-abslog3absx/2
- abs(2-(abs(log3abs(x divide by 2))))
-
Similar expressions
- abs(2+(abs(log3abs(x/2))))
Graphing y = abs(2-(abs(log3abs(x/2))))
The solution
You have entered
[src]
| | / |x|\||
f(x) = |2 - |log|3*|-||||
| | \ |2|/||
$$f{\left(x \right)} = \left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right|$$
f = Abs(2 - Abs(log(3*|x/2|)))
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:
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 4.9260373992871$$
$$x_{2} = -4.9260373992871$$
so we need to solve the equation:
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 4.9260373992871$$
$$x_{2} = -4.9260373992871$$
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
$$- \frac{\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} = 0$$
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
$$- \frac{\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} = 0$$
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(\frac{2 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)}^{2} \delta\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2\right) \operatorname{sign}^{3}{\left(x \right)} \operatorname{sign}^{3}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} + 4 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \delta\left(x\right) \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} + \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} + 2 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} \frac{d}{d x} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + \frac{\left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} - \frac{\left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} + \frac{x \frac{d}{d x} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}}{\operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} - 1\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x} - \frac{\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x}\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} = 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(\frac{2 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)}^{2} \delta\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2\right) \operatorname{sign}^{3}{\left(x \right)} \operatorname{sign}^{3}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}^{2}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} + 4 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \delta\left(x\right) \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} + \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \frac{d}{d x} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} + 2 \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)} \frac{d}{d x} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} + \frac{\left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} - 1\right) \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} - \frac{\left(\frac{2 x \delta\left(x\right)}{\operatorname{sign}{\left(x \right)}} + \frac{x \frac{d}{d x} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}}{\operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} - 1\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x} - \frac{\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}} \right)} \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)} \operatorname{sign}{\left(\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2 \right)} \operatorname{sign}{\left(\log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)} \right)}}{x}\right) \operatorname{sign}{\left(x \right)} \operatorname{sign}{\left(\frac{x}{\operatorname{sign}{\left(x \right)}} \right)}}{x \log{\left(\frac{3 x}{2 \operatorname{sign}{\left(x \right)}} \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \infty$$
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 Abs(2 - Abs(log(3*|x/2|))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right|}{x}\right) = 0$$
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:
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \left|{\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2}\right|$$
- No
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = - \left|{\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = \left|{\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2}\right|$$
- No
$$\left|{- \left|{\log{\left(3 \left|{\frac{x}{2}}\right| \right)}}\right| + 2}\right| = - \left|{\left|{\log{\left(\frac{3 \left|{x}\right|}{2} \right)}}\right| - 2}\right|$$
- No
so, the function
not is
neither even, nor odd
The graph