Mister Exam
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-
How to use it?
- Graphing y =:
- x+16
- x-1/2x+1
- sqrt(|x|)
- -sqrt(x-3)
- Limit of the function:
-
sqrt(|x|)
-
Identical expressions
- sqrt(|x|)
- square root of ( module of x|)
- √(|x|)
- sqrt|x|
-
Similar expressions
- \sqrt(|x|-1)+(2/(x^2-1))
- ((log(1-x)/log(2)))^(1/3)+(tan(1+1/x))/(sqrt((|x|))-2)
Graphing y = sqrt(|x|)
The solution
You have entered
[src]
_____ f(x) = \/ |x|
$$f{\left(x \right)} = \sqrt{\left|{x}\right|}$$
f = sqrt(|x|)
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:
$$\sqrt{\left|{x}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\sqrt{\left|{x}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 0$$
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
$$\frac{\operatorname{sign}{\left(x \right)}}{2 \sqrt{\left|{x}\right|}} = 0$$
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
$$\frac{\operatorname{sign}{\left(x \right)}}{2 \sqrt{\left|{x}\right|}} = 0$$
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
$$\frac{\delta\left(x\right) - \frac{\operatorname{sign}^{2}{\left(x \right)}}{4 \left|{x}\right|}}{\sqrt{\left|{x}\right|}} = 0$$
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
$$\frac{\delta\left(x\right) - \frac{\operatorname{sign}^{2}{\left(x \right)}}{4 \left|{x}\right|}}{\sqrt{\left|{x}\right|}} = 0$$
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
$$\lim_{x \to -\infty} \sqrt{\left|{x}\right|} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left|{x}\right|} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{\left|{x}\right|} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left|{x}\right|} = \infty$$
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 sqrt(|x|), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left|{x}\right|}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left|{x}\right|}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left|{x}\right|}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left|{x}\right|}}{x}\right) = 0$$
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:
$$\sqrt{\left|{x}\right|} = \sqrt{\left|{x}\right|}$$
- Yes
$$\sqrt{\left|{x}\right|} = - \sqrt{\left|{x}\right|}$$
- No
so, the function
is
even
So, check:
$$\sqrt{\left|{x}\right|} = \sqrt{\left|{x}\right|}$$
- Yes
$$\sqrt{\left|{x}\right|} = - \sqrt{\left|{x}\right|}$$
- No
so, the function
is
even