The points at which the function is not precisely defined: x1=−3 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: x2+2x−3x=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 x/(x^2 + 2*x - 1*3). (−1)3+02+2⋅00 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 (x2+2x−3)2x(−2x−2)+x2+2x−31=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+2x−3)22(x(x2+2x−34(x+1)2−1)−2x−2)=0 Solve this equation The roots of this equation x1=−332+33 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=−3 x2=1
x→−3−lim(x2+2x−3)22(x(x2+2x−34(x+1)2−1)−2x−2)=−∞ Let's take the limit x→−3+lim(x2+2x−3)22(x(x2+2x−34(x+1)2−1)−2x−2)=∞ Let's take the limit - the limits are not equal, so x1=−3 - is an inflection point x→1−lim(x2+2x−3)22(x(x2+2x−34(x+1)2−1)−2x−2)=−∞ Let's take the limit x→1+lim(x2+2x−3)22(x(x2+2x−34(x+1)2−1)−2x−2)=∞ 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 (−∞,−332+33] Convex at the intervals [−332+33,∞)
Vertical asymptotes
Have: x1=−3 x2=1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞lim(x2+2x−3x)=0 Let's take the limit so, equation of the horizontal asymptote on the left: y=0 x→∞lim(x2+2x−3x)=0 Let's take the limit so, equation of the horizontal asymptote on the right: y=0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(x^2 + 2*x - 1*3), divided by x at x->+oo and x ->-oo x→−∞limx2+2x−31=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞limx2+2x−31=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: x2+2x−3x=−x2−2x−3x - No x2+2x−3x=x2−2x−3x - No so, the function not is neither even, nor odd