Mister Exam
Graphing y = x(x-1)/((x+2)*(x-1))
The solution
You have entered
[src]
x*(x - 1)
f(x) = ---------------
(x + 2)*(x - 1)
$$f{\left(x \right)} = \frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)}$$
f = (x*(x - 1))/(((x - 1)*(x + 2)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -2$$
$$x_{2} = 1$$
$$x_{1} = -2$$
$$x_{2} = 1$$
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:
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = 0$$
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:
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = 0$$
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
$$\frac{x \left(- 2 x - 1\right)}{\left(x - 1\right) \left(x + 2\right)^{2}} + \frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(2 x - 1\right) = 0$$
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
$$\frac{x \left(- 2 x - 1\right)}{\left(x - 1\right) \left(x + 2\right)^{2}} + \frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(2 x - 1\right) = 0$$
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
$$\frac{\frac{x \left(\left(2 x + 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 1}\right) - 2 + \frac{2 x + 1}{x + 2} + \frac{2 x + 1}{x - 1}\right)}{x + 2} + 2 - \frac{2 \left(2 x - 1\right) \left(2 x + 1\right)}{\left(x - 1\right) \left(x + 2\right)}}{\left(x - 1\right) \left(x + 2\right)} = 0$$
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
$$\frac{\frac{x \left(\left(2 x + 1\right) \left(\frac{1}{x + 2} + \frac{1}{x - 1}\right) - 2 + \frac{2 x + 1}{x + 2} + \frac{2 x + 1}{x - 1}\right)}{x + 2} + 2 - \frac{2 \left(2 x - 1\right) \left(2 x + 1\right)}{\left(x - 1\right) \left(x + 2\right)}}{\left(x - 1\right) \left(x + 2\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -2$$
$$x_{2} = 1$$
$$x_{1} = -2$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x*(x - 1))/(((x + 2)*(x - 1))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(x - 1\right)\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(x - 1\right)\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(x - 1\right)\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1}{\left(x - 1\right) \left(x + 2\right)} \left(x - 1\right)\right) = 0$$
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:
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = - \frac{x}{2 - x}$$
- No
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = \frac{x}{2 - x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = - \frac{x}{2 - x}$$
- No
$$\frac{x \left(x - 1\right)}{\left(x - 1\right) \left(x + 2\right)} = \frac{x}{2 - x}$$
- No
so, the function
not is
neither even, nor odd