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-
How to use it?
- Graphing y =:
- (x^2-3x+2)/(x+1)
- x^2+4/2x
- x^2-4|x|-x
- x/(x-1)^2
- Derivative of:
-
(x^2-9*x)/(x^2-3*x)
- Limit of the function:
-
(x^2-9*x)/(x^2-3*x)
-
Identical expressions
- (x^ two - nine *x)/(x^ two - three *x)
- (x squared minus 9 multiply by x) divide by (x squared minus 3 multiply by x)
- (x to the power of two minus nine multiply by x) divide by (x to the power of two minus three multiply by x)
- (x2-9*x)/(x2-3*x)
- x2-9*x/x2-3*x
- (x²-9*x)/(x²-3*x)
- (x to the power of 2-9*x)/(x to the power of 2-3*x)
- (x^2-9x)/(x^2-3x)
- (x2-9x)/(x2-3x)
- x2-9x/x2-3x
- x^2-9x/x^2-3x
- (x^2-9*x) divide by (x^2-3*x)
-
Similar expressions
- (x^2+9*x)/(x^2-3*x)
- (x^2-9*x)/(x^2+3*x)
Graphing y = (x^2-9*x)/(x^2-3*x)
The solution
You have entered
[src]
2
x - 9*x
f(x) = --------
2
x - 3*x
$$f{\left(x \right)} = \frac{x^{2} - 9 x}{x^{2} - 3 x}$$
f = (x^2 - 9*x)/(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:
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 9$$
Numerical solution
$$x_{1} = 9$$
so we need to solve the equation:
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 9$$
Numerical solution
$$x_{1} = 9$$
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
$$\frac{\left(3 - 2 x\right) \left(x^{2} - 9 x\right)}{\left(x^{2} - 3 x\right)^{2}} + \frac{2 x - 9}{x^{2} - 3 x} = 0$$
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
$$\frac{\left(3 - 2 x\right) \left(x^{2} - 9 x\right)}{\left(x^{2} - 3 x\right)^{2}} + \frac{2 x - 9}{x^{2} - 3 x} = 0$$
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
$$\frac{2 \left(- \frac{\left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \left(x - 9\right)}{x - 3} + 1 - \frac{\left(2 x - 9\right) \left(2 x - 3\right)}{x \left(x - 3\right)}\right)}{x \left(x - 3\right)} = 0$$
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
$$\frac{2 \left(- \frac{\left(1 - \frac{\left(2 x - 3\right)^{2}}{x \left(x - 3\right)}\right) \left(x - 9\right)}{x - 3} + 1 - \frac{\left(2 x - 9\right) \left(2 x - 3\right)}{x \left(x - 3\right)}\right)}{x \left(x - 3\right)} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 9 x}{x^{2} - 3 x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{2} - 9 x}{x^{2} - 3 x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 9 x}{x^{2} - 3 x}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 1$$
$$\lim_{x \to \infty}\left(\frac{x^{2} - 9 x}{x^{2} - 3 x}\right) = 1$$
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^2 - 9*x)/(x^2 - 3*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 9 x}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} - 9 x}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 9 x}{x \left(x^{2} - 3 x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} - 9 x}{x \left(x^{2} - 3 x\right)}\right) = 0$$
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:
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = \frac{x^{2} + 9 x}{x^{2} + 3 x}$$
- No
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = - \frac{x^{2} + 9 x}{x^{2} + 3 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = \frac{x^{2} + 9 x}{x^{2} + 3 x}$$
- No
$$\frac{x^{2} - 9 x}{x^{2} - 3 x} = - \frac{x^{2} + 9 x}{x^{2} + 3 x}$$
- No
so, the function
not is
neither even, nor odd
The graph