Mister Exam
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-
How to use it?
- Graphing y =:
- -x(x-2)^2
- x|x-1|
- x^5+x
- |x|+5
- Derivative of:
-
(x+3)/(x-3)
- Integral of d{x}:
-
(x+3)/(x-3)
-
Identical expressions
- (x+ three)/(x- three)
- (x plus 3) divide by (x minus 3)
- (x plus three) divide by (x minus three)
- x+3/x-3
- (x+3) divide by (x-3)
-
Similar expressions
- f=2*ln((x+3)/x)-3
- 2ln*((x+3)/x)-3
- 2*log(x+3/x)-3
- (2lnx/x((x+3)/x))-3
- (2x-6)/(x-3)-((x^2-6x+3)/(x-3)^2)
- (x-3)/(x-3)
- (x+3)/(x+3)
Graphing y = (x+3)/(x-3)
The solution
You have entered
[src]
x + 3
f(x) = -----
x - 3
$$f{\left(x \right)} = \frac{x + 3}{x - 3}$$
f = (x + 3)/(x - 3)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 3$$
$$x_{1} = 3$$
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:
$$\frac{x + 3}{x - 3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
Numerical solution
$$x_{1} = -3$$
so we need to solve the equation:
$$\frac{x + 3}{x - 3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
Numerical solution
$$x_{1} = -3$$
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
$$\frac{1}{x - 3} - \frac{x + 3}{\left(x - 3\right)^{2}} = 0$$
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
$$\frac{1}{x - 3} - \frac{x + 3}{\left(x - 3\right)^{2}} = 0$$
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
$$\frac{2 \left(-1 + \frac{x + 3}{x - 3}\right)}{\left(x - 3\right)^{2}} = 0$$
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
$$\frac{2 \left(-1 + \frac{x + 3}{x - 3}\right)}{\left(x - 3\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 3$$
$$x_{1} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x - 3}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x + 3}{x - 3}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x - 3}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x + 3}{x - 3}\right) = 1$$
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 + 3)/(x - 3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x \left(x - 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + 3}{x \left(x - 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x \left(x - 3\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + 3}{x \left(x - 3\right)}\right) = 0$$
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:
$$\frac{x + 3}{x - 3} = \frac{3 - x}{- x - 3}$$
- No
$$\frac{x + 3}{x - 3} = - \frac{3 - x}{- x - 3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x + 3}{x - 3} = \frac{3 - x}{- x - 3}$$
- No
$$\frac{x + 3}{x - 3} = - \frac{3 - x}{- x - 3}$$
- No
so, the function
not is
neither even, nor odd