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-
How to use it?
- Graphing y =:
- -x^2+6x-3
- x^2-2x+8
- (x+5)^2
- x^4+4
-
Identical expressions
- (6x- eighteen)/(x^ two - three *x)
- (6x minus 18) divide by (x squared minus 3 multiply by x)
- (6x minus eighteen) divide by (x to the power of two minus three multiply by x)
- (6x-18)/(x2-3*x)
- 6x-18/x2-3*x
- (6x-18)/(x²-3*x)
- (6x-18)/(x to the power of 2-3*x)
- (6x-18)/(x^2-3x)
- (6x-18)/(x2-3x)
- 6x-18/x2-3x
- 6x-18/x^2-3x
- (6x-18) divide by (x^2-3*x)
-
Similar expressions
- (6x-18)/(x^2+3*x)
- (6x+18)/(x^2-3*x)
Graphing y = (6x-18)/(x^2-3*x)
The solution
You have entered
[src]
6*x - 18
f(x) = --------
2
x - 3*x
$$f{\left(x \right)} = \frac{6 x - 18}{x^{2} - 3 x}$$
f = (6*x - 1*18)/(x^2 - 3*x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
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:
$$\frac{6 x - 18}{x^{2} - 3 x} = 0$$
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:
$$\frac{6 x - 18}{x^{2} - 3 x} = 0$$
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
$$\frac{\left(- 2 x + 3\right) \left(6 x - 18\right)}{\left(x^{2} - 3 x\right)^{2}} + \frac{6}{x^{2} - 3 x} = 0$$
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
$$\frac{\left(- 2 x + 3\right) \left(6 x - 18\right)}{\left(x^{2} - 3 x\right)^{2}} + \frac{6}{x^{2} - 3 x} = 0$$
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
$$- \frac{12 \cdot \left(1 + \frac{2 x - 3}{x - 3} - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right)}{x^{2} \left(x - 3\right)} = 0$$
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
$$- \frac{12 \cdot \left(1 + \frac{2 x - 3}{x - 3} - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right)}{x^{2} \left(x - 3\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{6 x - 18}{x^{2} - 3 x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{6 x - 18}{x^{2} - 3 x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{6 x - 18}{x^{2} - 3 x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{6 x - 18}{x^{2} - 3 x}\right) = 0$$
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 (6*x - 1*18)/(x^2 - 3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{6 x - 18}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{6 x - 18}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{6 x - 18}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{6 x - 18}{x \left(x^{2} - 3 x\right)}\right) = 0$$
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:
$$\frac{6 x - 18}{x^{2} - 3 x} = \frac{- 6 x - 18}{x^{2} + 3 x}$$
- No
$$\frac{6 x - 18}{x^{2} - 3 x} = - \frac{- 6 x - 18}{x^{2} + 3 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{6 x - 18}{x^{2} - 3 x} = \frac{- 6 x - 18}{x^{2} + 3 x}$$
- No
$$\frac{6 x - 18}{x^{2} - 3 x} = - \frac{- 6 x - 18}{x^{2} + 3 x}$$
- No
so, the function
not is
neither even, nor odd
The graph