Mister Exam
Graphing y = √((x-2)/(3-x))
The solution
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_______
/ x - 2
f(x) = / -----
\/ 3 - x
$$f{\left(x \right)} = \sqrt{\frac{x - 2}{3 - x}}$$
f = sqrt((x - 2)/(3 - x))
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:
$$\sqrt{\frac{x - 2}{3 - x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
so we need to solve the equation:
$$\sqrt{\frac{x - 2}{3 - x}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
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
$$\frac{\sqrt{\frac{x - 2}{3 - x}} \left(3 - x\right) \left(\frac{1}{2 \left(3 - x\right)} + \frac{x - 2}{2 \left(3 - x\right)^{2}}\right)}{x - 2} = 0$$
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
$$\frac{\sqrt{\frac{x - 2}{3 - x}} \left(3 - x\right) \left(\frac{1}{2 \left(3 - x\right)} + \frac{x - 2}{2 \left(3 - x\right)^{2}}\right)}{x - 2} = 0$$
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
$$\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{9}{4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 3$$
$$\lim_{x \to 3^-}\left(\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)}\right) = \infty i$$
- 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:
Concave at the intervals
$$\left[\frac{9}{4}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{9}{4}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{9}{4}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 3$$
$$\lim_{x \to 3^-}\left(\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{\sqrt{- \frac{x - 2}{x - 3}} \left(1 - \frac{x - 2}{x - 3}\right) \left(\frac{1 - \frac{x - 2}{x - 3}}{x - 2} - \frac{2}{x - 2} - \frac{2}{x - 3}\right)}{4 \left(x - 2\right)}\right) = \infty i$$
- 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:
Concave at the intervals
$$\left[\frac{9}{4}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \frac{9}{4}\right]$$
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
$$\lim_{x \to -\infty} \sqrt{\frac{x - 2}{3 - x}} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \sqrt{\frac{x - 2}{3 - x}} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = i$$
$$\lim_{x \to -\infty} \sqrt{\frac{x - 2}{3 - x}} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = i$$
$$\lim_{x \to \infty} \sqrt{\frac{x - 2}{3 - x}} = i$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = i$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt((x - 2)/(3 - x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\frac{x - 2}{3 - x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\frac{x - 2}{3 - x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\frac{x - 2}{3 - x}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\frac{x - 2}{3 - x}}}{x}\right) = 0$$
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:
$$\sqrt{\frac{x - 2}{3 - x}} = \sqrt{\frac{- x - 2}{x + 3}}$$
- No
$$\sqrt{\frac{x - 2}{3 - x}} = - \sqrt{\frac{- x - 2}{x + 3}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\frac{x - 2}{3 - x}} = \sqrt{\frac{- x - 2}{x + 3}}$$
- No
$$\sqrt{\frac{x - 2}{3 - x}} = - \sqrt{\frac{- x - 2}{x + 3}}$$
- No
so, the function
not is
neither even, nor odd