Mister Exam
Graphing y = sqrt(y^2+1)/y
The solution
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f(y) = -----------
y
$$f{\left(y \right)} = \frac{\sqrt{y^{2} + 1}}{y}$$
f = sqrt(y^2 + 1)/y
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$y_{1} = 0$$
$$y_{1} = 0$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
$$\frac{\sqrt{y^{2} + 1}}{y} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y
so we need to solve the equation:
$$\frac{\sqrt{y^{2} + 1}}{y} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$\frac{1}{\sqrt{y^{2} + 1}} - \frac{\sqrt{y^{2} + 1}}{y^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$\frac{1}{\sqrt{y^{2} + 1}} - \frac{\sqrt{y^{2} + 1}}{y^{2}} = 0$$
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
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\frac{- \frac{\frac{y^{2}}{y^{2} + 1} - 1}{\sqrt{y^{2} + 1}} - \frac{2}{\sqrt{y^{2} + 1}} + \frac{2 \sqrt{y^{2} + 1}}{y^{2}}}{y} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$\frac{- \frac{\frac{y^{2}}{y^{2} + 1} - 1}{\sqrt{y^{2} + 1}} - \frac{2}{\sqrt{y^{2} + 1}} + \frac{2 \sqrt{y^{2} + 1}}{y^{2}}}{y} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$y_{1} = 0$$
$$y_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
$$\lim_{y \to -\infty}\left(\frac{\sqrt{y^{2} + 1}}{y}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{y \to \infty}\left(\frac{\sqrt{y^{2} + 1}}{y}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{y \to -\infty}\left(\frac{\sqrt{y^{2} + 1}}{y}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{y \to \infty}\left(\frac{\sqrt{y^{2} + 1}}{y}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(y^2 + 1)/y, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty}\left(\frac{\sqrt{y^{2} + 1}}{y^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(\frac{\sqrt{y^{2} + 1}}{y^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{y \to -\infty}\left(\frac{\sqrt{y^{2} + 1}}{y^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(\frac{\sqrt{y^{2} + 1}}{y^{2}}\right) = 0$$
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(-y) и f = -f(-y).
So, check:
$$\frac{\sqrt{y^{2} + 1}}{y} = - \frac{\sqrt{y^{2} + 1}}{y}$$
- No
$$\frac{\sqrt{y^{2} + 1}}{y} = \frac{\sqrt{y^{2} + 1}}{y}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sqrt{y^{2} + 1}}{y} = - \frac{\sqrt{y^{2} + 1}}{y}$$
- No
$$\frac{\sqrt{y^{2} + 1}}{y} = \frac{\sqrt{y^{2} + 1}}{y}$$
- No
so, the function
not is
neither even, nor odd