Mister Exam
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-
How to use it?
- Graphing y =:
- x^2+2x-8
- x³/(x-2)²
- x^3-9x
-
x/(1-x)
- Derivative of:
-
x/(1-x)
- Integral of d{x}:
-
x/(1-x)
- Limit of the function:
-
x/(1-x)
-
Identical expressions
- x/(one -x)
- x divide by (1 minus x)
- x divide by (one minus x)
- x/1-x
- x divide by (1-x)
-
Similar expressions
- exp(x)/(1-x*exp(x)+exp(x))
- ln(1+x)/1-x
- x/(1-x^2)
- sgrt(x)/1-x^2
- 2*x/(1-x^2)
- x/(1+x)
Graphing y = x/(1-x)
The solution
You have entered
[src]
x
f(x) = -----
1 - x
$$f{\left(x \right)} = \frac{x}{- x + 1}$$
f = x/(1 - x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\frac{x}{- x + 1} = 0$$
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:
$$\frac{x}{- x + 1} = 0$$
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
$$\frac{x}{\left(- x + 1\right)^{2}} + \frac{1}{- x + 1} = 0$$
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
$$\frac{x}{\left(- x + 1\right)^{2}} + \frac{1}{- x + 1} = 0$$
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
$$\frac{2 \left(- \frac{x}{x - 1} + 1\right)}{\left(x - 1\right)^{2}} = 0$$
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
$$\frac{2 \left(- \frac{x}{x - 1} + 1\right)}{\left(x - 1\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 1$$
$$x_{1} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{- x + 1}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{- x + 1}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(\frac{x}{- x + 1}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{x}{- x + 1}\right) = -1$$
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/(1 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{- x + 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{- x + 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{- x + 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{- x + 1} = 0$$
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:
$$\frac{x}{- x + 1} = - \frac{x}{x + 1}$$
- No
$$\frac{x}{- x + 1} = \frac{x}{x + 1}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x}{- x + 1} = - \frac{x}{x + 1}$$
- No
$$\frac{x}{- x + 1} = \frac{x}{x + 1}$$
- No
so, the function
not is
neither even, nor odd
The graph