The points at which the function is not precisely defined: y1=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: yy2+1=0 Solve this equation Solution is not found, it's possible that the graph doesn't intersect the axis Y
The points of intersection with the Y axis coordinate
The graph crosses Y axis when y equals 0: substitute y = 0 to sqrt(y^2 + 1)/y. 002+1 The result: f(0)=∞~ sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation dydf(y)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dydf(y)= the first derivative y2+11−y2y2+1=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 dy2d2f(y)=0 (the second derivative equals zero), the roots of this equation will be the inflection points for the specified function graph: dy2d2f(y)= the second derivative y−y2+1y2+1y2−1−y2+12+y22y2+1=0 Solve this equation Solutions are not found, maybe, the function has no inflections
Vertical asymptotes
Have: y1=0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo y→−∞lim(yy2+1)=−1 Let's take the limit so, equation of the horizontal asymptote on the left: y=−1 y→∞lim(yy2+1)=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 y→−∞lim(y2y2+1)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right y→∞lim(y2y2+1)=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: yy2+1=−yy2+1 - No yy2+1=yy2+1 - No so, the function not is neither even, nor odd