Decide if the following pair of functions are of the same order.

(a) f(x) = x^2 - 7 and g(x) = x^2 + 7

first question: do you know what the order of a function is? The term "order" has several meanings.

Do you maybe mean degree of a polynomial?

If so, then it's the highest power of the variable.

I'm not sure.

eecs.yorku.ca/course_archive/2015-16/F/1019/assn/A6.pdf

Question 30 shows an example but I don't really understand.

To determine if the two functions, f(x) = x^2 - 7 and g(x) = x^2 + 7, are of the same order, we need to compare their growth rates as x approaches infinity.

Let's start by finding the limits of the functions as x approaches infinity:

Limit of f(x) as x approaches infinity:
lim(x -> ∞) f(x) = lim(x -> ∞) (x^2 - 7)
Since the term x^2 dominates as x approaches infinity, the constant term (-7) becomes insignificant. Therefore, we can simplify the equation to:
lim(x -> ∞) f(x) = lim(x -> ∞) x^2

Limit of g(x) as x approaches infinity:
lim(x -> ∞) g(x) = lim(x -> ∞) (x^2 + 7)
Again, as x approaches infinity, the constant term (7) becomes insignificant, and we can simplify the equation to:
lim(x -> ∞) g(x) = lim(x -> ∞) x^2

Since the limits of both f(x) and g(x) as x approaches infinity are the same (both equal to x^2), we can conclude that the two functions, f(x) = x^2 - 7 and g(x) = x^2 + 7, are of the same order.

In general, when comparing the order of two functions, we look at the dominant terms. If the dominant terms are the same or have the same power, we say the functions are of the same order.