Btw, Lonewolf meant |x| [>=] 7 (the absolute value of x must be equal to or greater than 7), since at x = 7 or x = -7, the function value is [squ]0 = 0.
Another way of saying "|x| [>=] 7" is "x [<=] -7 or 7 [>=] 7"
Btw, I'll try to give an explanation of a function's natural domain. If it is unclear or wrong please point me out.
--- natural domain of a function ---
[note that we are discussing a function that maps a real number to another real number]
The domain of a function can be stated explicitly, such as
f(x) = 1, x >0
The domain of the function above is all real numbers greater than 0. With that restriction, f(0), f(-2), f(-3.43), etc are undefined.
If a function has no domain stated explicitly, then it is understood that the domain consists of all real number x for which f(x) is defined. This implicit domain is called the natural domain of f(x).
Let's look at the function f(x) = 1/x
for x = 1, f(x) = 1/1 = 1 (defined). For x = 5, f(x) = 1/5 (defined also). It turns out that only at x = 0 is f(x) undefined. That is because 1/0 is undefined. So the domain of the function f(x) = 1/x is all x values other than 0.
How about the function f(x)=[squ]x?
At x equal or greater than 0, f(x) is defined. At x less than 0, f(x) is undefined. For example, [squ](-1) is undefined (because there are no real number y that satisfies the property y^2 = -1). That means the natural domain of the function f(x) = [squ]x is all x equal to or greater than 0.
Now, let's go back to your problem. Your function is f(x)=[squ](x^2-49). Let's introduce a new variable b, where b = x^2-49. We can then rewrite your function as f(x) = [squ]b. We already know from the previous explanation that in order to make f(x) defined, b must be equal to or greater than 0. So we write
b [>=] 0
But b = x^2-49, so we can write
x^2-49 [>=] 0
By factoring, we get
(x+7)(x-7) [>=] 0
At x = 7 or x = -7, the value of b = (x+7)(x-7) will be zero (do you know why?). Since [squ]0 is defined, f(x) is defined at these x = 7 and x = -7. At values of x between 7 and -7, the value of b will be less than 0 (verify this). That means f(x) is undefined for -7 < x < 7. For x < -7 or x > 7, b is positive (verify this also) and that means f(x) is defined.
Thus we can conclude that the natural domain of f(x) consists of all x such that
x is equal to -7, or
x is smaller than -7, or
x is equal to 7, or
x is greater than 7
We can write this as |x| [>=] 7.
Does every part of my explanation makes sense to you?
