Mister Exam
Graphing y = (x^2+6x+3)/(x+4)
The solution
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2
x + 6*x + 3
f(x) = ------------
x + 4
$$f{\left(x \right)} = \frac{\left(x^{2} + 6 x\right) + 3}{x + 4}$$
f = (x^2 + 6*x + 3)/(x + 4)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -4$$
$$x_{1} = -4$$
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:
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 - \sqrt{6}$$
$$x_{2} = -3 + \sqrt{6}$$
Numerical solution
$$x_{1} = -0.550510257216822$$
$$x_{2} = -5.44948974278318$$
so we need to solve the equation:
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3 - \sqrt{6}$$
$$x_{2} = -3 + \sqrt{6}$$
Numerical solution
$$x_{1} = -0.550510257216822$$
$$x_{2} = -5.44948974278318$$
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
$$\frac{2 x + 6}{x + 4} - \frac{\left(x^{2} + 6 x\right) + 3}{\left(x + 4\right)^{2}} = 0$$
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
$$\frac{2 x + 6}{x + 4} - \frac{\left(x^{2} + 6 x\right) + 3}{\left(x + 4\right)^{2}} = 0$$
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
$$\frac{2 \left(- \frac{2 \left(x + 3\right)}{x + 4} + 1 + \frac{x^{2} + 6 x + 3}{\left(x + 4\right)^{2}}\right)}{x + 4} = 0$$
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
$$\frac{2 \left(- \frac{2 \left(x + 3\right)}{x + 4} + 1 + \frac{x^{2} + 6 x + 3}{\left(x + 4\right)^{2}}\right)}{x + 4} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -4$$
$$x_{1} = -4$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x + 4}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x + 4}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x + 4}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x + 4}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x^2 + 6*x + 3)/(x + 4), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x \left(x + 4\right)}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x \left(x + 4\right)}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x \left(x + 4\right)}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(x^{2} + 6 x\right) + 3}{x \left(x + 4\right)}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = \frac{x^{2} - 6 x + 3}{4 - x}$$
- No
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = - \frac{x^{2} - 6 x + 3}{4 - x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = \frac{x^{2} - 6 x + 3}{4 - x}$$
- No
$$\frac{\left(x^{2} + 6 x\right) + 3}{x + 4} = - \frac{x^{2} - 6 x + 3}{4 - x}$$
- No
so, the function
not is
neither even, nor odd