scope of this text. The Fig . gives a sketch of y = f ( x ) = x , y = f ( x ) = x , y = f ( x ) = x and y = f ( x ) = x . Observe that the curves get steeper as the power of x increases.
Steeper the curve, faster is the rate of growth. What this means is that for a fixed increment in the value of x (> ), the increment in the value of y = f n ( x ) increases as n increases for n = , , , . It is conceivable that such a statement is true for all positive values of n , where f n ( x ) = x n . Essentially, this means that the graph of y = f n ( x ) leans more towards the y -axis as n increases.
For example, consider f ( x ) = x and f ( x ) = x . If x increases from to , f increases from to whereas f increases from to . Thus, for the same increment in x , f grow faster than f . Upshot of the above discussion is that the growth of polynomial functions is dependent on the degree of the polynomial function – higher the degree, greater is the growth.
The next natural question is: Is there a function which grows faster than any polynomial function. The answer is in affirmative and an example of such a function is y = f ( x ) = x . Our claim is that this function f grows faster than f n ( x ) = x n for any positive integer n . For example, we can prove that x grows faster than f ( x ) = x .
For large values of x like x = ,