Mister Exam
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-
How to use it?
- Graphing y =:
- |x|-1/|x|-x^2
- y=(9x+1)/(9x^2+x)
- 3tgx
-
sqrt(arctan(sqrt(2x+3)))
-
Identical expressions
- y=(9x+ one)/(9x^ two +x)
- y equally (9x plus 1) divide by (9x squared plus x)
- y equally (9x plus one) divide by (9x to the power of two plus x)
- y=(9x+1)/(9x2+x)
- y=9x+1/9x2+x
- y=(9x+1)/(9x²+x)
- y=(9x+1)/(9x to the power of 2+x)
- y=9x+1/9x^2+x
- y=(9x+1) divide by (9x^2+x)
-
Similar expressions
- y=(9x-1)/(9x^2+x)
- y=(9x+1)/(9x^2-x)
Graphing y = y=(9x+1)/(9x^2+x)
The solution
You have entered
[src]
9*x + 1
f(x) = --------
2
9*x + x
$$f{\left(x \right)} = \frac{9 x + 1}{9 x^{2} + x}$$
f = (9*x + 1)/(9*x^2 + x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -0.111111111111111$$
$$x_{2} = 0$$
$$x_{1} = -0.111111111111111$$
$$x_{2} = 0$$
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:
$$\frac{9 x + 1}{9 x^{2} + x} = 0$$
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:
$$\frac{9 x + 1}{9 x^{2} + x} = 0$$
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
$$\frac{\left(- 18 x - 1\right) \left(9 x + 1\right)}{\left(9 x^{2} + x\right)^{2}} + \frac{9}{9 x^{2} + x} = 0$$
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
$$\frac{\left(- 18 x - 1\right) \left(9 x + 1\right)}{\left(9 x^{2} + x\right)^{2}} + \frac{9}{9 x^{2} + x} = 0$$
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
$$- \frac{2 \left(9 + \frac{9 \left(18 x + 1\right)}{9 x + 1} - \frac{\left(18 x + 1\right)^{2}}{x \left(9 x + 1\right)}\right)}{x^{2} \left(9 x + 1\right)} = 0$$
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
$$- \frac{2 \left(9 + \frac{9 \left(18 x + 1\right)}{9 x + 1} - \frac{\left(18 x + 1\right)^{2}}{x \left(9 x + 1\right)}\right)}{x^{2} \left(9 x + 1\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -0.111111111111111$$
$$x_{2} = 0$$
$$x_{1} = -0.111111111111111$$
$$x_{2} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{9 x + 1}{9 x^{2} + x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{9 x + 1}{9 x^{2} + x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{9 x + 1}{9 x^{2} + x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{9 x + 1}{9 x^{2} + x}\right) = 0$$
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 (9*x + 1)/(9*x^2 + x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{9 x + 1}{x \left(9 x^{2} + x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{9 x + 1}{x \left(9 x^{2} + x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{9 x + 1}{x \left(9 x^{2} + x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{9 x + 1}{x \left(9 x^{2} + x\right)}\right) = 0$$
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:
$$\frac{9 x + 1}{9 x^{2} + x} = \frac{1 - 9 x}{9 x^{2} - x}$$
- No
$$\frac{9 x + 1}{9 x^{2} + x} = - \frac{1 - 9 x}{9 x^{2} - x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{9 x + 1}{9 x^{2} + x} = \frac{1 - 9 x}{9 x^{2} - x}$$
- No
$$\frac{9 x + 1}{9 x^{2} + x} = - \frac{1 - 9 x}{9 x^{2} - x}$$
- No
so, the function
not is
neither even, nor odd