The points at which the function is not precisely defined: x1=−1 x2=1
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: atan(−xx+1x)=0 Solve this equation The points of intersection with 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 atan(x/(sqrt(1 - x*x))). atan(−0⋅0+10) The result: f(0)=0 The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation dxdf(x)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dxdf(x)= the first derivative −xx+1x2+1(−xx+1)23x2+−xx+11=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 dx2d2f(x)=0 (the second derivative equals zero), the roots of this equation will be the inflection points for the specified function graph: dx2d2f(x)= the second derivative −−x2+1(x2−1x2−1)x(−x2+1x2+1)(x2−12+−x2+13)=0 Solve this equation The roots of this equation x1=0 You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function: Points where there is an indetermination: x1=−1 x2=1
x→−1−lim−−x2+1(x2−1x2−1)x(−x2+1x2+1)(x2−12+−x2+13)=−∞i Let's take the limit x→−1+lim−−x2+1(x2−1x2−1)x(−x2+1x2+1)(x2−12+−x2+13)=−∞ Let's take the limit - the limits are not equal, so x1=−1 - is an inflection point x→1−lim−−x2+1(x2−1x2−1)x(−x2+1x2+1)(x2−12+−x2+13)=∞ Let's take the limit x→1+lim−−x2+1(x2−1x2−1)x(−x2+1x2+1)(x2−12+−x2+13)=∞i Let's take the limit - the limits are not equal, so x2=1 - is an inflection point
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 [0,∞) Convex at the intervals (−∞,0]
Vertical asymptotes
Have: x1=−1 x2=1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞limatan(−xx+1x)=x→−∞limatan(−xx+1x) Let's take the limit so, equation of the horizontal asymptote on the left: y=x→−∞limatan(−xx+1x) x→∞limatan(−xx+1x)=x→∞limatan(−xx+1x) Let's take the limit so, equation of the horizontal asymptote on the right: y=x→∞limatan(−xx+1x)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(x/(sqrt(1 - x*x))), divided by x at x->+oo and x ->-oo x→−∞limxatan(−xx+1x)=x→−∞limxatan(−xx+1x) Let's take the limit so, inclined asymptote equation on the left: y=xx→−∞limxatan(−xx+1x) x→∞limxatan(−xx+1x)=x→∞limxatan(−xx+1x) Let's take the limit so, inclined asymptote equation on the right: y=xx→∞limxatan(−xx+1x)
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: atan(−xx+1x)=−atan(−x2+1x) - No atan(−xx+1x)=atan(−x2+1x) - No so, the function not is neither even, nor odd