Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x*e^(-(x)^2)
- -x^4+2x^3+2
-
-x^3+x^2-x+6
- sin(x)
-
Identical expressions
- (zero , twenty-five - zero ,5x)|x|\x- two
- (0,25 minus 0,5x) module of x|\x minus 2
- (zero , twenty minus five minus zero ,5x) module of x|\x minus two
- 0,25-0,5x|x|\x-2
-
Similar expressions
- (0,25+0,5x)|x|\x-2
- (0,25-0,5x)|x|\x+2
Graphing y = (0,25-0,5x)|x|\x-2
The solution
You have entered
[src]
/1 x\
|- - -|*|x|
\4 2/
f(x) = ----------- - 2
x
$$f{\left(x \right)} = -2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}$$
f = -2 + ((1/4 - x/2)*|x|)/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:
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{\left(\frac{1}{4} - \frac{x}{2}\right) \operatorname{sign}{\left(x \right)} - \frac{\left|{x}\right|}{2}}{x} - \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x^{2}} = 0$$
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
$$\frac{\left(\frac{1}{4} - \frac{x}{2}\right) \operatorname{sign}{\left(x \right)} - \frac{\left|{x}\right|}{2}}{x} - \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x^{2}} = 0$$
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
$$\frac{- \frac{\left(2 x - 1\right) \delta\left(x\right)}{2} - \operatorname{sign}{\left(x \right)} + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)}}{4 x} + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)} + 2 \left|{x}\right|}{4 x} + \frac{\left|{x}\right|}{2 x} - \frac{\left(2 x - 1\right) \left|{x}\right|}{2 x^{2}}}{x} = 0$$
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
$$\frac{- \frac{\left(2 x - 1\right) \delta\left(x\right)}{2} - \operatorname{sign}{\left(x \right)} + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)}}{4 x} + \frac{\left(2 x - 1\right) \operatorname{sign}{\left(x \right)} + 2 \left|{x}\right|}{4 x} + \frac{\left|{x}\right|}{2 x} - \frac{\left(2 x - 1\right) \left|{x}\right|}{2 x^{2}}}{x} = 0$$
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
$$\lim_{x \to -\infty}\left(-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}\right) = -\infty$$
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 ((1/4 - x/2)*|x|)/x - 2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
$$\lim_{x \to -\infty}\left(\frac{-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}}{x}\right) = \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{x}{2}$$
$$\lim_{x \to \infty}\left(\frac{-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x}}{x}\right) = - \frac{1}{2}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - \frac{x}{2}$$
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:
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = -2 - \frac{\left(\frac{x}{2} + \frac{1}{4}\right) \left|{x}\right|}{x}$$
- No
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = 2 + \frac{\left(\frac{x}{2} + \frac{1}{4}\right) \left|{x}\right|}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = -2 - \frac{\left(\frac{x}{2} + \frac{1}{4}\right) \left|{x}\right|}{x}$$
- No
$$-2 + \frac{\left(\frac{1}{4} - \frac{x}{2}\right) \left|{x}\right|}{x} = 2 + \frac{\left(\frac{x}{2} + \frac{1}{4}\right) \left|{x}\right|}{x}$$
- No
so, the function
not is
neither even, nor odd