Mister Exam
Graphing y = √(x+2)-(x+3)/(x+2)
The solution
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_______ x + 3
f(x) = \/ x + 2 - -----
x + 2
$$f{\left(x \right)} = \sqrt{x + 2} - \frac{x + 3}{x + 2}$$
f = sqrt(x + 2) - (x + 3)/(x + 2)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -2$$
$$x_{1} = -2$$
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:
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{5}{3} + \frac{7}{9 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{47}{54}}} + \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{47}{54}}$$
Numerical solution
$$x_{1} = 0.147899035704787$$
$$x_{2} = 0.147899035704778$$
$$x_{3} = 0.147899035704788$$
$$x_{4} = 0.14789903570482$$
so we need to solve the equation:
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{5}{3} + \frac{7}{9 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{47}{54}}} + \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{47}{54}}$$
Numerical solution
$$x_{1} = 0.147899035704787$$
$$x_{2} = 0.147899035704778$$
$$x_{3} = 0.147899035704788$$
$$x_{4} = 0.14789903570482$$
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
$$- \frac{- x - 3}{\left(x + 2\right)^{2}} - \frac{1}{x + 2} + \frac{1}{2 \sqrt{x + 2}} = 0$$
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
$$- \frac{- x - 3}{\left(x + 2\right)^{2}} - \frac{1}{x + 2} + \frac{1}{2 \sqrt{x + 2}} = 0$$
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
$$\frac{2}{\left(x + 2\right)^{2}} - \frac{2 \left(x + 3\right)}{\left(x + 2\right)^{3}} - \frac{1}{4 \left(x + 2\right)^{\frac{3}{2}}} = 0$$
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
$$\frac{2}{\left(x + 2\right)^{2}} - \frac{2 \left(x + 3\right)}{\left(x + 2\right)^{3}} - \frac{1}{4 \left(x + 2\right)^{\frac{3}{2}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -2$$
$$x_{1} = -2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sqrt{x + 2} - \frac{x + 3}{x + 2}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x + 2} - \frac{x + 3}{x + 2}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\sqrt{x + 2} - \frac{x + 3}{x + 2}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{x + 2} - \frac{x + 3}{x + 2}\right) = \infty$$
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 sqrt(x + 2) - (x + 3)/(x + 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 2} - \frac{x + 3}{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} - \frac{x + 3}{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x + 2} - \frac{x + 3}{x + 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{x + 2} - \frac{x + 3}{x + 2}}{x}\right) = 0$$
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:
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = \sqrt{2 - x} - \frac{3 - x}{2 - x}$$
- No
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = - \sqrt{2 - x} + \frac{3 - x}{2 - x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = \sqrt{2 - x} - \frac{3 - x}{2 - x}$$
- No
$$\sqrt{x + 2} - \frac{x + 3}{x + 2} = - \sqrt{2 - x} + \frac{3 - x}{2 - x}$$
- No
so, the function
not is
neither even, nor odd