Mister Exam
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-
How to use it?
- Graphing y =:
- x|x|
- √x+√(4-x)
-
x^3+3x^2-24x+28
- x^2(x-7)
-
Identical expressions
- (|x|+x- four)/x- two
- ( module of x| plus x minus 4) divide by x minus 2
- ( module of x| plus x minus four) divide by x minus two
- |x|+x-4/x-2
- (|x|+x-4) divide by x-2
-
Similar expressions
- (|x|+x+4)/x-2
- (|x|-x-4)/x-2
- (|x|+x-4)/x+2
Graphing y = (|x|+x-4)/x-2
The solution
You have entered
[src]
|x| + x - 4
f(x) = ----------- - 2
x
$$f{\left(x \right)} = -2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}$$
f = -2 + (x + |x| - 4)/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:
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
so we need to solve the equation:
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
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
$$\frac{\operatorname{sign}{\left(x \right)} + 1}{x} - \frac{\left(x + \left|{x}\right|\right) - 4}{x^{2}} = 0$$
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
$$\frac{\operatorname{sign}{\left(x \right)} + 1}{x} - \frac{\left(x + \left|{x}\right|\right) - 4}{x^{2}} = 0$$
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
$$\frac{2 \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)} + 1}{x} + \frac{x + \left|{x}\right| - 4}{x^{2}}\right)}{x} = 0$$
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
$$\frac{2 \left(\delta\left(x\right) - \frac{\operatorname{sign}{\left(x \right)} + 1}{x} + \frac{x + \left|{x}\right| - 4}{x^{2}}\right)}{x} = 0$$
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
$$\lim_{x \to -\infty}\left(-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}\right) = -2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -2$$
$$\lim_{x \to \infty}\left(-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}\right) = -2$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -2$$
$$\lim_{x \to \infty}\left(-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (|x| + x - 4)/x - 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x}}{x}\right) = 0$$
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:
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = -2 - \frac{- x + \left|{x}\right| - 4}{x}$$
- No
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = 2 + \frac{- x + \left|{x}\right| - 4}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = -2 - \frac{- x + \left|{x}\right| - 4}{x}$$
- No
$$-2 + \frac{\left(x + \left|{x}\right|\right) - 4}{x} = 2 + \frac{- x + \left|{x}\right| - 4}{x}$$
- No
so, the function
not is
neither even, nor odd