Mister Exam
Graphing y = xsqrt(x-1)^2/|x|
The solution
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2
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x*\/ x - 1
f(x) = ------------
|x|
$$f{\left(x \right)} = \frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}$$
f = (x*(sqrt(x - 1))^2)/|x|
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
so we need to solve the equation:
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 1$$
Numerical solution
$$x_{1} = 1$$
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
$$\frac{x + \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} - \frac{\left(x - 1\right) \operatorname{sign}{\left(x \right)}}{x} = 0$$
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
$$\frac{x + \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} - \frac{\left(x - 1\right) \operatorname{sign}{\left(x \right)}}{x} = 0$$
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
$$2 \left(\frac{1}{\left|{x}\right|} - \frac{\left(x - 1\right) \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)}}{x}\right)}{x} - \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)}}{x^{2}}\right) = 0$$
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
$$2 \left(\frac{1}{\left|{x}\right|} - \frac{\left(x - 1\right) \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)}}{x}\right)}{x} - \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|}\right) = \infty$$
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 (x*(sqrt(x - 1))^2)/|x|, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{\left|{x}\right|}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = - \frac{x \left(- x - 1\right)}{\left|{x}\right|}$$
- No
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = \frac{x \left(- x - 1\right)}{\left|{x}\right|}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = - \frac{x \left(- x - 1\right)}{\left|{x}\right|}$$
- No
$$\frac{x \left(\sqrt{x - 1}\right)^{2}}{\left|{x}\right|} = \frac{x \left(- x - 1\right)}{\left|{x}\right|}$$
- No
so, the function
not is
neither even, nor odd