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How to use it?
- Graphing y =:
- -x^2-4x-2
- x²-3x+1
- -x^2+2x-2
- 6x^2-9x-x^3
- Limit of the function:
-
(-16+x^2)/(4+2*x^2+9*x)
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Identical expressions
- (- sixteen +x^ two)/(four + two *x^ two + nine *x)
- ( minus 16 plus x squared ) divide by (4 plus 2 multiply by x squared plus 9 multiply by x)
- ( minus sixteen plus x to the power of two) divide by (four plus two multiply by x to the power of two plus nine multiply by x)
- (-16+x2)/(4+2*x2+9*x)
- -16+x2/4+2*x2+9*x
- (-16+x²)/(4+2*x²+9*x)
- (-16+x to the power of 2)/(4+2*x to the power of 2+9*x)
- (-16+x^2)/(4+2x^2+9x)
- (-16+x2)/(4+2x2+9x)
- -16+x2/4+2x2+9x
- -16+x^2/4+2x^2+9x
- (-16+x^2) divide by (4+2*x^2+9*x)
-
Similar expressions
- (-16+x^2)/(4-2*x^2+9*x)
- (-16-x^2)/(4+2*x^2+9*x)
- (-16+x^2)/(4+2*x^2-9*x)
- (16+x^2)/(4+2*x^2+9*x)
Graphing y = (-16+x^2)/(4+2*x^2+9*x)
The solution
You have entered
[src]
2
-16 + x
f(x) = --------------
2
4 + 2*x + 9*x
$$f{\left(x \right)} = \frac{x^{2} - 16}{2 x^{2} + 9 x + 4}$$
f = (x^2 - 16)/(2*x^2 + 9*x + 4)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -4$$
$$x_{2} = -0.5$$
$$x_{1} = -4$$
$$x_{2} = -0.5$$
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:
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 4$$
Numerical solution
$$x_{1} = 4$$
so we need to solve the equation:
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 4$$
Numerical solution
$$x_{1} = 4$$
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
$$\frac{2 x}{2 x^{2} + 9 x + 4} + \frac{\left(- 4 x - 9\right) \left(x^{2} - 16\right)}{\left(2 x^{2} + 9 x + 4\right)^{2}} = 0$$
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
$$\frac{2 x}{2 x^{2} + 9 x + 4} + \frac{\left(- 4 x - 9\right) \left(x^{2} - 16\right)}{\left(2 x^{2} + 9 x + 4\right)^{2}} = 0$$
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
$$\frac{2 \left(- \frac{2 x \left(4 x + 9\right)}{2 x^{2} + 9 x + 4} + \frac{\left(x^{2} - 16\right) \left(\frac{\left(4 x + 9\right)^{2}}{2 x^{2} + 9 x + 4} - 2\right)}{2 x^{2} + 9 x + 4} + 1\right)}{2 x^{2} + 9 x + 4} = 0$$
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
$$\frac{2 \left(- \frac{2 x \left(4 x + 9\right)}{2 x^{2} + 9 x + 4} + \frac{\left(x^{2} - 16\right) \left(\frac{\left(4 x + 9\right)^{2}}{2 x^{2} + 9 x + 4} - 2\right)}{2 x^{2} + 9 x + 4} + 1\right)}{2 x^{2} + 9 x + 4} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -4$$
$$x_{2} = -0.5$$
$$x_{1} = -4$$
$$x_{2} = -0.5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 16}{2 x^{2} + 9 x + 4}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x^{2} - 16}{2 x^{2} + 9 x + 4}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 16}{2 x^{2} + 9 x + 4}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x^{2} - 16}{2 x^{2} + 9 x + 4}\right) = \frac{1}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{1}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-16 + x^2)/(4 + 2*x^2 + 9*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 16}{x \left(2 x^{2} + 9 x + 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} - 16}{x \left(2 x^{2} + 9 x + 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x^{2} - 16}{x \left(2 x^{2} + 9 x + 4\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x^{2} - 16}{x \left(2 x^{2} + 9 x + 4\right)}\right) = 0$$
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:
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = \frac{x^{2} - 16}{2 x^{2} - 9 x + 4}$$
- No
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = - \frac{x^{2} - 16}{2 x^{2} - 9 x + 4}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = \frac{x^{2} - 16}{2 x^{2} - 9 x + 4}$$
- No
$$\frac{x^{2} - 16}{2 x^{2} + 9 x + 4} = - \frac{x^{2} - 16}{2 x^{2} - 9 x + 4}$$
- No
so, the function
not is
neither even, nor odd
The graph