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Graphing y = sqrt(3x-5)+sqrt(3x+11)

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         _________     __________
f(x) = \/ 3*x - 5  + \/ 3*x + 11

f{\left(x \right)} = \sqrt{3 x - 5} + \sqrt{3 x + 11}

f = sqrt(3*x - 5) + sqrt(3*x + 11)

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{3 x - 5} + \sqrt{3 x + 11} = 0

Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(3*x - 5) + sqrt(3*x + 11).

\sqrt{0 \cdot 3 + 11} + \sqrt{-5 + 0 \cdot 3}

The result:

f{\left(0 \right)} = \sqrt{11} + \sqrt{5} i

The point:

(0, sqrt(11) + i*sqrt(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)} =

\frac{3}{2 \sqrt{3 x + 11}} + \frac{3}{2 \sqrt{3 x - 5}} = 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{9 \left(\frac{1}{\left(3 x + 11\right)^{\frac{3}{2}}} + \frac{1}{\left(3 x - 5\right)^{\frac{3}{2}}}\right)}{4} = 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}\left(\sqrt{3 x - 5} + \sqrt{3 x + 11}\right) = \infty i

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(\sqrt{3 x - 5} + \sqrt{3 x + 11}\right) = \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(3*x - 5) + sqrt(3*x + 11), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{\sqrt{3 x - 5} + \sqrt{3 x + 11}}{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{3 x - 5} + \sqrt{3 x + 11}}{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{3 x - 5} + \sqrt{3 x + 11} = \sqrt{11 - 3 x} + \sqrt{- 3 x - 5}

- No

\sqrt{3 x - 5} + \sqrt{3 x + 11} = - \sqrt{11 - 3 x} - \sqrt{- 3 x - 5}

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