Mister Exam
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-
How to use it?
- Graphing y =:
- abs(x^2-2*abs(x)-3)
-
log0.5/x
-
|x|-4/x^2-4|x|
- ((4+x)^2)/(4*(2+x))
-
Identical expressions
- abs(x^ two - two *abs(x)- three)
- abs(x squared minus 2 multiply by abs(x) minus 3)
- abs(x to the power of two minus two multiply by abs(x) minus three)
- abs(x2-2*abs(x)-3)
- absx2-2*absx-3
- abs(x²-2*abs(x)-3)
- abs(x to the power of 2-2*abs(x)-3)
- abs(x^2-2abs(x)-3)
- abs(x2-2abs(x)-3)
- absx2-2absx-3
- absx^2-2absx-3
-
Similar expressions
- abs(x^2+2*abs(x)-3)
- abs(x^2-2*abs(x)+3)
Graphing y = abs(x^2-2*abs(x)-3)
The solution
You have entered
[src]
| 2 | f(x) = |x - 2*|x| - 3|
$$f{\left(x \right)} = \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|$$
f = Abs(x^2 - 2*|x| - 3)
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:
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
$$x_{2} = 3$$
Numerical solution
$$x_{1} = 3$$
$$x_{2} = -3$$
so we need to solve the equation:
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
$$x_{2} = 3$$
Numerical solution
$$x_{1} = 3$$
$$x_{2} = -3$$
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
$$2 \left(4 \left(x - \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(- x^{2} + 2 \left|{x}\right| + 3\right) + \left(2 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(- x^{2} + 2 \left|{x}\right| + 3 \right)}\right) = 0$$
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
$$2 \left(4 \left(x - \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(- x^{2} + 2 \left|{x}\right| + 3\right) + \left(2 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(- x^{2} + 2 \left|{x}\right| + 3 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \infty$$
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(x^2 - 2*|x| - 3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|$$
- Yes
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = - \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|$$
- No
so, the function
is
even
So, check:
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|$$
- Yes
$$\left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right| = - \left|{\left(x^{2} - 2 \left|{x}\right|\right) - 3}\right|$$
- No
so, the function
is
even