Question on Domain/Range terminology.?

Question

I have to find the maximal domain for the following: 
Cos(x) / ((x^2) - 9) 

Which is all real numbers except 3 and -3 
How do I write this (assuming it's correct?) 

A second question that popped up while I was trying to figure this out. what does |x| mean? 

thanks!

Answer 1

1).

The function is f(x) = cos(x) / (x² - 9).

The function is a rational function.

We know that all possible values of x  is domain of a function.

A rational function is simply fraction and in a fraction the denominator cannot be equal to 0 because it would be undefined.

To find which number make the fraction undefined create an equation where the denominator is not equal to zero.

x² - 9 ≠ 0

Add 9 to each side.

x² ≠ 9

x² ≠ ( ± 3)²

⇒ x ≠ - 3 and x ≠ 3.

Domain is {x ∈ R : x ≠ - 3 and x ≠ 3 }.

Yes, your assumption is correct.

2).

|x| is just saying absolute value of x, which means the distance of x from zero (it can't be negative).