Suppose we have functions
![](https://latex.codecogs.com/png.latex?\bg_white y=f(x))
and
![](https://latex.codecogs.com/png.latex?\bg_white y=g(x))
where
![](https://latex.codecogs.com/png.latex?\bg_white g(x)=\sqrt{f(x)})
.
Vertical asymptotes of
![](https://latex.codecogs.com/png.latex?\bg_white y=f(x))
could occur when
![](https://latex.codecogs.com/png.latex?\bg_white f(x))
is undefined.
Vertical asymptotes on
![](https://latex.codecogs.com/png.latex?\bg_white y=g(x)=\sqrt{f(x)})
mean looking for where
![](https://latex.codecogs.com/png.latex?\bg_white g(x))
is undefined, and that means considering both
- anywhere that
(as outside the domain of
)
- as well as anywhere that
is undefined.
Boundaries of undefined areas (where
![](https://latex.codecogs.com/png.latex?\bg_white f(x)=0)
) will become end points rather than asymptotes as
![](https://latex.codecogs.com/png.latex?\bg_white \sqrt{0} =0)
and is thus well-defined.
Horizontal asymptotes are found by examining the beahaviour of a function as a
![](https://latex.codecogs.com/png.latex?\bg_white x \to \pm\infty)
.
In your specific case:
![](https://latex.codecogs.com/png.latex?\bg_white f(x) = 2 + \cfrac{1}{x+2} \qquad \qquad \text{and} \qquad \qquad g(x) = \sqrt{f(x)} = \sqrt{2 +\cfrac{1}{x+2}})
As every real constant except zero has a reciprocal, it is clear that
![](https://latex.codecogs.com/png.latex?\bg_white f(x))
is only undefined if
![](https://latex.codecogs.com/png.latex?\bg_white x+2=0)
and thus the domain of
![](https://latex.codecogs.com/png.latex?\bg_white f(x)\ \text{is}\ \left\{x \in \mathbb{R},\ x \ne -2\right\})
and that
![](https://latex.codecogs.com/png.latex?\bg_white x=-2)
is a vertical asymptote because:
![](https://latex.codecogs.com/png.latex?\bg_white \text{As}\ x \to -2^-,\ x+2 \to 0^-,\ \text{and so}\ \cfrac{1}{x+2} \to -\infty.\ \text{Thus,}\ f(x) \to -\infty.)
And, the result is a typical rectangular hyperbola with the asymptotes shifted to
![](https://latex.codecogs.com/png.latex?\bg_white x = -2)
and
![](https://latex.codecogs.com/png.latex?\bg_white y = 2)
(shown as green dotted lines).
Now, for
![](https://latex.codecogs.com/png.latex?\bg_white y = g(x))
, the domain requires
![](https://latex.codecogs.com/png.latex?\bg_white f(x) \geqslant 0)
and so the part of
![](https://latex.codecogs.com/png.latex?\bg_white f(x))
below the
![](https://latex.codecogs.com/png.latex?\bg_white x)
-axis will be outside the domain of
![](https://latex.codecogs.com/png.latex?\bg_white g(x))
. The domain of
![](https://latex.codecogs.com/png.latex?\bg_white g(x))
is thus:
![](https://latex.codecogs.com/png.latex?\bg_white \left\{x \in \mathbb{R},\ x > -2\ \text{or}\ x \leqslant -\cfrac{5}{2}\right\})
The
![](https://latex.codecogs.com/png.latex?\bg_white x)
-intercept of
![](https://latex.codecogs.com/png.latex?\bg_white y=f(x))
, at
![](https://latex.codecogs.com/png.latex?\bg_white x=-2.5)
, will remain a zero for
![](https://latex.codecogs.com/png.latex?\bg_white y=g(x))
but it also becomes an end point, and one with a vertical tangent as
![](https://latex.codecogs.com/png.latex?\bg_white \begin{align*} g(x) = \sqrt{f(x)} =\big[f(x)\big]^{\frac{1}{2}} \qquad \implies \qquad g'(x) &= \cfrac{1}{2}\big[f(x)\big]^{-\frac{1}{2}} \times f'(x) \qquad \text{(Chain Rule)} \\ &= \cfrac{f'(x)}{2\sqrt{f(x)}} \qquad \text{which is undefined for $f(x)=0$ provided $f'(x) \ne 0$} \end{align*})
There will remain a vertical asymptote at
![](https://latex.codecogs.com/png.latex?\bg_white x = -2)
as
![](https://latex.codecogs.com/png.latex?\bg_white g(x))
is undefined there. The horizontal asymptote will shift to
![](https://latex.codecogs.com/png.latex?\bg_white y = \sqrt{2})
as
![](https://latex.codecogs.com/png.latex?\bg_white f(x) \to 2 \implies g(x) = \sqrt{f(x)} \to \sqrt{2})
.
The curve
![](https://latex.codecogs.com/png.latex?\bg_white y = f(x))
will lie above
![](https://latex.codecogs.com/png.latex?\bg_white y=g(x))
wherever
![](https://latex.codecogs.com/png.latex?\bg_white f(x) > 1)
as the square root of any value greater than 1 produces a smaller number (for example,
![](https://latex.codecogs.com/png.latex?\bg_white \sqrt{2} = 1.414... < 2)
).
The two curves will meet only when
![](https://latex.codecogs.com/png.latex?\bg_white f(x) = 0)
of
![](https://latex.codecogs.com/png.latex?\bg_white f(x) = 1)
as 0 and 1 are the only two numbers that are there own square root. The intersections are thus
![](https://latex.codecogs.com/png.latex?\bg_white \big(-2.5,\ 0\big))
and
![](https://latex.codecogs.com/png.latex?\bg_white \big(-3,\ 1\big))
.
The curve
![](https://latex.codecogs.com/png.latex?\bg_white y = f(x))
will lie below
![](https://latex.codecogs.com/png.latex?\bg_white y=g(x))
wherever
![](https://latex.codecogs.com/png.latex?\bg_white 0 < f(x) < 1)
as the square root of any positive value less than 1 produces a greater number (for example,
![](https://latex.codecogs.com/png.latex?\bg_white \sqrt{0.25} = 0.5 > 0.25)
).
The results from plotting the curves is shown below, matching the above description. The new asymptote at
![](https://latex.codecogs.com/png.latex?\bg_white y = \sqrt{2})
has been added as a purple dotted line.
![BoS - hyperbola and its square root.png](/data/attachments/32/32338-d1fdd48c20f6c7acae94cfc2d48ed2c1.jpg)