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Graphing y = sqrt((|2x-3|)-2)

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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

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(|2*x - 3| - 2).

\sqrt{-2 + \left|{-3 + 0 \cdot 2}\right|}

The result:

f{\left(0 \right)} = 1

The point:

(0, 1)

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)} =

\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)} =

\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

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

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