Graphing y = sqrt(tan(x))

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f(x) = \/ tan(x)

f{\left(x \right)} = \sqrt{\tan{\left(x \right)}}

f = sqrt(tan(x))

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{\tan{\left(x \right)}} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = 0

Numerical solution

x_{1} = 0

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(tan(x)).

\sqrt{\tan{\left(0 \right)}}

The result:

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

The point:

(0, 0)

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{\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}}{\sqrt{\tan{\left(x \right)}}} = 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

\left(4 \sqrt{\tan{\left(x \right)}} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}}\right) \left(\frac{\tan^{2}{\left(x \right)}}{4} + \frac{1}{4}\right) = 0

Solve this equation
The roots of this equation

x_{1} = - \frac{\pi}{6}

x_{2} = \frac{\pi}{6}

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{\pi}{6}, \infty\right)

Convex at the intervals

\left(-\infty, \frac{\pi}{6}\right]

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{\tan{\left(x \right)}} = \lim_{x \to -\infty} \sqrt{\tan{\left(x \right)}}

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = \lim_{x \to -\infty} \sqrt{\tan{\left(x \right)}}

\lim_{x \to \infty} \sqrt{\tan{\left(x \right)}} = \lim_{x \to \infty} \sqrt{\tan{\left(x \right)}}

Let's take the limit
so,
equation of the horizontal asymptote on the right:

y = \lim_{x \to \infty} \sqrt{\tan{\left(x \right)}}

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of sqrt(tan(x)), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right)

Let's take the limit
so,
inclined asymptote equation on the left:

y = x \lim_{x \to -\infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right)

\lim_{x \to \infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right) = \lim_{x \to \infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right)

Let's take the limit
so,
inclined asymptote equation on the right:

y = x \lim_{x \to \infty}\left(\frac{\sqrt{\tan{\left(x \right)}}}{x}\right)

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{\tan{\left(x \right)}} = \sqrt{- \tan{\left(x \right)}}

- No

\sqrt{\tan{\left(x \right)}} = - \sqrt{- \tan{\left(x \right)}}

- No
so, the function
not is
neither even, nor odd

The graph

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Graphing y = sqrt(tan(x))

The graph:

Intersection points:

Piecewise:

The solution