Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x/(1-x^2)
  • x^2-2x+3
  • x/(3-x^2)
  • x^2+6x+5
  • Identical expressions

  • |x^ two - five *(|x|)+ six |- two
  • module of x squared minus 5 multiply by (|x|) plus 6| minus 2
  • module of x to the power of two minus five multiply by (|x|) plus six | minus two
  • |x2-5*(|x|)+6|-2
  • |x2-5*|x|+6|-2
  • |x²-5*(|x|)+6|-2
  • |x to the power of 2-5*(|x|)+6|-2
  • |x^2-5(|x|)+6|-2
  • |x2-5(|x|)+6|-2
  • |x2-5|x|+6|-2
  • |x^2-5|x|+6|-2
  • Similar expressions

  • |x^2+5*(|x|)+6|-2
  • |x^2-5*(|x|)+6|+2
  • |x^2-5*(|x|)-6|-2

Graphing y = |x^2-5*(|x|)+6|-2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       | 2            |    
f(x) = |x  - 5*|x| + 6| - 2
$$f{\left(x \right)} = \left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2$$
f = Abs(x^2 - 5*|x| + 6) - 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:
$$\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2 = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = -4$$
$$x_{2} = -1$$
$$x_{3} = 1$$
$$x_{4} = 4$$
Numerical solution
$$x_{1} = 1$$
$$x_{2} = 4$$
$$x_{3} = -1$$
$$x_{4} = -4$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(x^2 - 5*|x| + 6) - 2.
$$-2 + \left|{\left(0^{2} - 5 \left|{0}\right|\right) + 6}\right|$$
The result:
$$f{\left(0 \right)} = 4$$
The point:
(0, 4)
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
$$2 \left(\left(2 x - 5 \operatorname{sign}{\left(x \right)}\right)^{2} \delta\left(x^{2} - 5 \left|{x}\right| + 6\right) - \left(5 \delta\left(x\right) - 1\right) \operatorname{sign}{\left(x^{2} - 5 \left|{x}\right| + 6 \right)}\right) = 0$$
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
$$\lim_{x \to -\infty}\left(\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2\right) = \infty$$
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(x^2 - 5*|x| + 6) - 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2}{x}\right) = \infty$$
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:
$$\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2 = \left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2$$
- Yes
$$\left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right| - 2 = 2 - \left|{\left(x^{2} - 5 \left|{x}\right|\right) + 6}\right|$$
- No
so, the function
is
even