The progression of a function

When figuring out the progression of a function, we try to find out as much as possible about the behavior of the function. We are interested in such things as monotonicity, i.e. whether the function is increasing or decreasing, or on what intervals the function is increasing or decreasing. We are also interested in the extremes of the function - the minima and maxima.

Procedure

At the beginning we are given a function f, typically by some rule like f(x) = x². Our task is to find out as much information about this function as possible. Typically we find out:

1. Thedefinitional domain of the function and the domain of values of the function.
2. Determine whether the function is even or odd
3. We find out if the function is bounded.
4. Calculate the intercepts with the x axis and with the y axis. These are easily calculated by putting a zero after x and calculating f(0). This gives us the intersection with the axis y. The intersections with the axis x are obtained by setting the entire function equal to zero f(x) = 0 and finding all the roots of the equation.
5. We find the extremes of the function and find the monotonicity of the function.
6. Find the inflection points and the intervals of convexity and concavity