Show that each of these pairs of functions are of the same order.a) 3x + 7, x________________b) 2x2 + x ? 7, x2________________c) ?x + 1/2?, x2________________d) log(x2 + 1), log2x________________e) log10x. log2x
Solution: Step 1:In this problem, we have to show that each of given functions is of the same order.Step 2: The definition for Big- Omega:Let f is Big - Omega of g, written if there are positive constant C and k such that for x > k.Big-Omega is very similar to big-O. Big -notation is used to indicate a lower bound on the functions for the large value of the independent variable.The definition for Big- O: Let f and g be functions from the real numbers to the real numbers. Then f is O(g) if there are constants c and kSuch that Step 3:(a):Let f(x) = 3x + 7 and g(x) = xIt is clear that the functions 3x + 7 and x are polynomials of degree 1.So, x < 3x + 7 for x > 7It follows that That is 3x + 7 is And also, 3x + 7 4x for x > 7Therefore Thus, 3x + 7 is O(x).Hence 3x +7 and x are of same order.Step 4:(b):Let f(x) = 2x2 + x -7 and g(x) = x2It is clear that the functions 2x2 + x -7 and x2 are polynomials of degree 2.So, x2 < 2x2 + x -7 for x 7It follows that That is 2x2 + x -7 is And also, 2x2 + x -7 3x2 for x 1Therefore Thus, 2x2 + x -7 is O(x).Hence 2x2 + x -7 and x2 are of the same order.