Mister Exam
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-
How to use it?
- Graphing y =:
- x^3+x^2-x+1
- (x^2-x-6)/(x-2)
- x^2-6x+10
- x^2+5x+6
- Derivative of:
-
(x+3)/(x^2-9)
-
Identical expressions
- (x+ three)/(x^ two - nine)
- (x plus 3) divide by (x squared minus 9)
- (x plus three) divide by (x to the power of two minus nine)
- (x+3)/(x2-9)
- x+3/x2-9
- (x+3)/(x²-9)
- (x+3)/(x to the power of 2-9)
- x+3/x^2-9
- (x+3) divide by (x^2-9)
-
Similar expressions
- (x+3)/(x^2+9)
- (x-3)/(x^2-9)
Graphing y = (x+3)/(x^2-9)
The solution
You have entered
[src]
x + 3
f(x) = ------
2
x - 9
$$f{\left(x \right)} = \frac{x + 3}{x^{2} - 9}$$
f = (x + 3)/(x^2 - 1*9)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$x_{1} = -3$$
$$x_{2} = 3$$
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:
$$\frac{x + 3}{x^{2} - 9} = 0$$
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:
$$\frac{x + 3}{x^{2} - 9} = 0$$
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
$$- \frac{2 x \left(x + 3\right)}{\left(x^{2} - 9\right)^{2}} + \frac{1}{x^{2} - 9} = 0$$
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
$$- \frac{2 x \left(x + 3\right)}{\left(x^{2} - 9\right)^{2}} + \frac{1}{x^{2} - 9} = 0$$
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
$$\frac{2 \left(\left(x + 3\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right) - 2 x\right)}{\left(x^{2} - 9\right)^{2}} = 0$$
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
$$\frac{2 \left(\left(x + 3\right) \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right) - 2 x\right)}{\left(x^{2} - 9\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$x_{1} = -3$$
$$x_{2} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x^{2} - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x + 3}{x^{2} - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x^{2} - 9}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x + 3}{x^{2} - 9}\right) = 0$$
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 (x + 3)/(x^2 - 1*9), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x \left(x^{2} - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + 3}{x \left(x^{2} - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x + 3}{x \left(x^{2} - 9\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x + 3}{x \left(x^{2} - 9\right)}\right) = 0$$
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:
$$\frac{x + 3}{x^{2} - 9} = \frac{- x + 3}{x^{2} - 9}$$
- No
$$\frac{x + 3}{x^{2} - 9} = - \frac{- x + 3}{x^{2} - 9}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x + 3}{x^{2} - 9} = \frac{- x + 3}{x^{2} - 9}$$
- No
$$\frac{x + 3}{x^{2} - 9} = - \frac{- x + 3}{x^{2} - 9}$$
- No
so, the function
not is
neither even, nor odd
The graph