Mister Exam
Graphing y = sqrt((3x-2)/(2x+6))
The solution
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_________
/ 3*x - 2
f(x) = / -------
\/ 2*x + 6
$$f{\left(x \right)} = \sqrt{\frac{3 x - 2}{2 x + 6}}$$
f = sqrt((3*x - 2)/(2*x + 6))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -3$$
$$x_{1} = -3$$
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:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2}{3}$$
Numerical solution
$$x_{1} = 0.666666666666667$$
so we need to solve the equation:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{2}{3}$$
Numerical solution
$$x_{1} = 0.666666666666667$$
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
$$\frac{\sqrt{\frac{3 x - 2}{2 x + 6}} \left(2 x + 6\right) \left(\frac{3}{2 \left(2 x + 6\right)} - \frac{3 x - 2}{\left(2 x + 6\right)^{2}}\right)}{3 x - 2} = 0$$
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
$$\frac{\sqrt{\frac{3 x - 2}{2 x + 6}} \left(2 x + 6\right) \left(\frac{3}{2 \left(2 x + 6\right)} - \frac{3 x - 2}{\left(2 x + 6\right)^{2}}\right)}{3 x - 2} = 0$$
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
$$\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -3$$
$$\lim_{x \to -3^-}\left(\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)}\right) = \infty$$
$$\lim_{x \to -3^+}\left(\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -3$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -3$$
$$\lim_{x \to -3^-}\left(\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)}\right) = \infty$$
$$\lim_{x \to -3^+}\left(\frac{\sqrt{2} \sqrt{\frac{3 x - 2}{x + 3}} \left(3 - \frac{3 x - 2}{x + 3}\right) \left(\frac{3 - \frac{3 x - 2}{x + 3}}{3 x - 2} - \frac{6}{3 x - 2} - \frac{2}{x + 3}\right)}{8 \left(3 x - 2\right)}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -3$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = -3$$
$$x_{1} = -3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\frac{3 x - 2}{2 x + 6}} = \frac{\sqrt{6}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\sqrt{6}}{2}$$
$$\lim_{x \to \infty} \sqrt{\frac{3 x - 2}{2 x + 6}} = \frac{\sqrt{6}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\sqrt{6}}{2}$$
$$\lim_{x \to -\infty} \sqrt{\frac{3 x - 2}{2 x + 6}} = \frac{\sqrt{6}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\sqrt{6}}{2}$$
$$\lim_{x \to \infty} \sqrt{\frac{3 x - 2}{2 x + 6}} = \frac{\sqrt{6}}{2}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\sqrt{6}}{2}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt((3*x - 2)/(2*x + 6)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\frac{3 x - 2}{2 x + 6}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\frac{3 x - 2}{2 x + 6}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\frac{3 x - 2}{2 x + 6}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\frac{3 x - 2}{2 x + 6}}}{x}\right) = 0$$
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:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = \sqrt{\frac{- 3 x - 2}{6 - 2 x}}$$
- No
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = - \sqrt{\frac{- 3 x - 2}{6 - 2 x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = \sqrt{\frac{- 3 x - 2}{6 - 2 x}}$$
- No
$$\sqrt{\frac{3 x - 2}{2 x + 6}} = - \sqrt{\frac{- 3 x - 2}{6 - 2 x}}$$
- No
so, the function
not is
neither even, nor odd