I don't understand the end behavior of a polynomial function or INFINITY?

Question

I don't understand infinity at all, can you please explain the end behavior of x squared, x to the third.

Answer 1

Because you haven't set a "limit" to how far the function y=x^2 can go in the left and right directions, its x-values can go all the way to infinity (i.e.: there will be points for the graphs x^2 and x^3 to the x-value of infinity)

Answer 2

Infinity is a tricky concept. Note first that it is not a number, and so there is no "end" to these functions at all -- they just keep going. Second note that even simpler polynomial functions like f(x)=x head to infinity, albeit more slowly.

The point of infinity (from the Latin "infinitum" meaning "not finite" or "without limit") is that you cannot give me a number that is the highest possible number. I can always add 1 to it (or double it or whatever) to get a higher number. There is no boundary or limit to, say, the set of all integers.

Likewise, all polynomials that exist over any and all values of x (the number of values that x can take on is unbounded or infinite) will go to either positive or negative infinity.

Some functions restrict the domain of the function so that x is only valid for some particular range, e.g.

f(x) = {x, if x is in the range [-10, 10]; and 0 otherwise}.

Other functions that are not polynomials with a finite number of terms may oscillate back and forth never going to infinity in either direction (e.g., sin and cosine functions), while still others may approach an asymptote, as in g(x)=2/(e^x+e^-x), which is known as the hyperbolic secant function and which approaches 0 as x gets very large in either the positive or negative direction.