Mister Exam
Graphing y = sqrt((|2x-3|)-2)
The solution
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_______________ f(x) = \/ |2*x - 3| - 2
$$f{\left(x \right)} = \sqrt{\left|{2 x - 3}\right| - 2}$$
f = sqrt(|2*x - 3| - 2)
The graph of the function
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:
$$\sqrt{\left|{2 x - 3}\right| - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
$$x_{2} = \frac{5}{2}$$
Numerical solution
$$x_{1} = 0.5$$
$$x_{2} = 2.5$$
so we need to solve the equation:
$$\sqrt{\left|{2 x - 3}\right| - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{2}$$
$$x_{2} = \frac{5}{2}$$
Numerical solution
$$x_{1} = 0.5$$
$$x_{2} = 2.5$$
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
$$\frac{\operatorname{sign}{\left(2 x - 3 \right)}}{\sqrt{\left|{2 x - 3}\right| - 2}} = 0$$
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
$$\frac{\operatorname{sign}{\left(2 x - 3 \right)}}{\sqrt{\left|{2 x - 3}\right| - 2}} = 0$$
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
$$\frac{4 \delta\left(2 x - 3\right) - \frac{\operatorname{sign}^{2}{\left(2 x - 3 \right)}}{\left|{2 x - 3}\right| - 2}}{\sqrt{\left|{2 x - 3}\right| - 2}} = 0$$
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
$$\frac{4 \delta\left(2 x - 3\right) - \frac{\operatorname{sign}^{2}{\left(2 x - 3 \right)}}{\left|{2 x - 3}\right| - 2}}{\sqrt{\left|{2 x - 3}\right| - 2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\left|{2 x - 3}\right| - 2} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left|{2 x - 3}\right| - 2} = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sqrt{\left|{2 x - 3}\right| - 2} = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sqrt{\left|{2 x - 3}\right| - 2} = \infty$$
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(|2*x - 3| - 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left|{2 x - 3}\right| - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left|{2 x - 3}\right| - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left|{2 x - 3}\right| - 2}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left|{2 x - 3}\right| - 2}}{x}\right) = 0$$
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:
$$\sqrt{\left|{2 x - 3}\right| - 2} = \sqrt{\left|{2 x + 3}\right| - 2}$$
- No
$$\sqrt{\left|{2 x - 3}\right| - 2} = - \sqrt{\left|{2 x + 3}\right| - 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\left|{2 x - 3}\right| - 2} = \sqrt{\left|{2 x + 3}\right| - 2}$$
- No
$$\sqrt{\left|{2 x - 3}\right| - 2} = - \sqrt{\left|{2 x + 3}\right| - 2}$$
- No
so, the function
not is
neither even, nor odd