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

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

I found this example solution online:
a). f(x) = 3x + 7 and g(x) = x
x <= 3x + 7 <= 4x for all x > 7

I need help making sense of this so I can apply it to my work. Thanks.

We can't do your homework for you. Sorry.

I was never asking for the answer

To determine whether two functions are of the same order, we typically compare the growth rates of the functions as their input values get larger and larger. One common way to do this is to use the limit as x approaches infinity.

In the given example, we are comparing f(x) = x^2 - 7 and g(x) = x^2 + 7. To determine if they are of the same order, we will compare their growth rates using limits.

To start, let's simplify each function to their leading term. The leading term is the term with the highest exponent.

For f(x) = x^2 - 7, the leading term is x^2.
For g(x) = x^2 + 7, the leading term is also x^2.

Now, let's determine the limit as x approaches infinity for the ratio of these leading terms:

lim(x->∞) [(x^2 - 7)/(x^2 + 7)]

To simplify this limit, we can divide both the numerator and denominator by x^2:

lim(x->∞) [(1 - 7/x^2)/(1 + 7/x^2)]

As x approaches infinity, 7/x^2 approaches 0. So, the limit becomes:

lim(x->∞) [(1 - 7(0))/(1 + 7(0))] = lim(x->∞) (1/1) = 1

Since the limit is equal to 1, this means that the growth rates of f(x) and g(x) are similar as x approaches infinity. Hence, we can conclude that f(x) and g(x) are of the same order.

Now, let's interpret the example you mentioned: f(x) = 3x + 7 and g(x) = x. The manipulation you see is to compare these functions by finding an inequality relation between them. This can be helpful in some cases to determine the order of growth.

For this example, they compared the two functions by using the inequality:

x ≤ 3x + 7 ≤ 4x for all x > 7

This means that for any value of x greater than 7, the value of 3x + 7 will be between x and 4x. In other words, the difference between 3x + 7 and x is never negative, and the difference between 3x + 7 and 4x is also never negative.

Using this inequality relation, you can infer that the growth rate of f(x) is somewhere between the growth rate of x and 4x as x becomes larger.

I hope this explanation helps you understand how to compare functions and determine their order of growth. If you have any further questions, feel free to ask!