Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x⁴-1
- |x+3|-|x|
- -x^3+x
- (x-3)/(x-4)
- Limit of the function:
-
(-9+3*sqrt(x))/(9-x)
-
Identical expressions
- (- nine + three *sqrt(x))/(nine -x)
- ( minus 9 plus 3 multiply by square root of (x)) divide by (9 minus x)
- ( minus nine plus three multiply by square root of (x)) divide by (nine minus x)
- (-9+3*√(x))/(9-x)
- (-9+3sqrt(x))/(9-x)
- -9+3sqrtx/9-x
- (-9+3*sqrt(x)) divide by (9-x)
-
Similar expressions
- (9+3*sqrt(x))/(9-x)
- (-9-3*sqrt(x))/(9-x)
- (-9+3*sqrt(x))/(9+x)
Graphing y = (-9+3*sqrt(x))/(9-x)
The solution
You have entered
[src]
___
-9 + 3*\/ x
f(x) = ------------
9 - x
$$f{\left(x \right)} = \frac{3 \sqrt{x} - 9}{9 - x}$$
f = (3*sqrt(x) - 9)/(9 - x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 9$$
$$x_{1} = 9$$
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:
$$\frac{3 \sqrt{x} - 9}{9 - x} = 0$$
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:
$$\frac{3 \sqrt{x} - 9}{9 - x} = 0$$
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
$$\frac{3 \sqrt{x} - 9}{\left(9 - x\right)^{2}} + \frac{3}{2 \sqrt{x} \left(9 - x\right)} = 0$$
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
$$\frac{3 \sqrt{x} - 9}{\left(9 - x\right)^{2}} + \frac{3}{2 \sqrt{x} \left(9 - x\right)} = 0$$
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
$$\frac{3 \left(- \frac{2 \left(\sqrt{x} - 3\right)}{\left(x - 9\right)^{2}} + \frac{1}{\sqrt{x} \left(x - 9\right)} + \frac{1}{4 x^{\frac{3}{2}}}\right)}{x - 9} = 0$$
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
$$\frac{3 \left(- \frac{2 \left(\sqrt{x} - 3\right)}{\left(x - 9\right)^{2}} + \frac{1}{\sqrt{x} \left(x - 9\right)} + \frac{1}{4 x^{\frac{3}{2}}}\right)}{x - 9} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 9$$
$$x_{1} = 9$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{3 \sqrt{x} - 9}{9 - x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{3 \sqrt{x} - 9}{9 - x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{3 \sqrt{x} - 9}{9 - x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{3 \sqrt{x} - 9}{9 - x}\right) = 0$$
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 + 3*sqrt(x))/(9 - x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{3 \sqrt{x} - 9}{x \left(9 - x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{3 \sqrt{x} - 9}{x \left(9 - x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{3 \sqrt{x} - 9}{x \left(9 - x\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{3 \sqrt{x} - 9}{x \left(9 - x\right)}\right) = 0$$
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:
$$\frac{3 \sqrt{x} - 9}{9 - x} = \frac{3 \sqrt{- x} - 9}{x + 9}$$
- No
$$\frac{3 \sqrt{x} - 9}{9 - x} = - \frac{3 \sqrt{- x} - 9}{x + 9}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{3 \sqrt{x} - 9}{9 - x} = \frac{3 \sqrt{- x} - 9}{x + 9}$$
- No
$$\frac{3 \sqrt{x} - 9}{9 - x} = - \frac{3 \sqrt{- x} - 9}{x + 9}$$
- No
so, the function
not is
neither even, nor odd