What actually growth of a function represents? I am going through the book Discrete Mathematics by Kenneth Rosen. In chapter 3, there is a topic "The Growth of Functions" and then Big-O Notation. I am really unable to understand what actually Growth of Functions represents? Kindly share some insights with examples. That would be of great help. Thank you so much for the help.
PS: I am new to this topic.

Growth of a particular function is basically how a function would behave for sufficiently large values of the input to function. You particularly ask about the big -O notation. 
Let us say we have two functions $f(x)$ and $g(x)$ and if we say that $f(x)=O(g(x))$ then that means that there exists  numbers $c$ and $n_0$ such that $f(x) \leq c.g(x) \forall x \geq n_0$.
So you basically are saying that for sufficiently large values of $x$ (which is depicted in condition $x\geq n_0$) the function $f(x)$ is below the function $c.g(x)$ for all further values. In a way you are predicting the behaviour of the function in terms of another function. Basically you are setting an upper bound for the function $f(x)$ in terms of $g(x)$.
For example let's take $f(x)=x$ and $g(x)=x^2$. Now, note that for all $x \geq 1$, $f(x)\leq 1.g(x)$. So, in this example you get $c=1 , n_0=1$. 
Note that $g(x)$ is indeed an upper bound but this upper bound is not tight. And you can get a better approximation.
So, generally when we talk about growth of functions we are interested in a very tight bound. 
There are many places where you can't exactly pinpoint the value of a function so you give a big - O bound for that function. For example, the great prime number theorem gives us the estimate of the number of primes up till a given number uses this notation.\
