Decide if the following pair of functions are of the same order. The functions f(x) and g(x) are of the same order if f(x) is O(g(x)) and f(x) is Ω(g(x).
(a) f(x) = x^2 - 7 and g(x) = x^2 + 7
To determine if the functions f(x) and g(x) are of the same order, we need to compare their growth rates. One way to do this is by analyzing the limit of their ratio as x approaches infinity.
Let's calculate the limit of f(x) / g(x) as x approaches infinity:
lim (x->∞) f(x) / g(x) = lim (x->∞) (x^2 - 7) / (x^2 + 7)
To simplify this expression, we can divide both the numerator and denominator by x^2:
= lim (x->∞) (1 - 7/x^2) / (1 + 7/x^2)
Now, as x approaches infinity, the term 7/x^2 tends to zero:
= (1 - 0) / (1 + 0)
= 1
The limit of f(x) / g(x) is equal to 1. This means that as x approaches infinity, the ratio of f(x) to g(x) approaches a constant value, which implies that f(x) and g(x) have the same growth rate.
Therefore, we can conclude that the functions f(x) = x^2 - 7 and g(x) = x^2 + 7 are of the same order.