Functions finding the domains

a) x+1 / √ x^2-4

b) 1/√ x^2-1

thank youuuu

If you think of a function as a "recipe" and the variable as the "ingredient", then the domain is the list of ALL the ingredients you could use.

More formally, the domain is the set of all values (for the variable) for which the function is "well-defined" (meaning = the function will give you one, and only one, clear answer for that value).

f(x) = (x+1) / √(x^2-4)

Here, you find out what values of x could cause trouble:

1) Because there is a division, you check to see if it is possible for the denominator (bottom) to be equal to zero. A division by zero is rarely "well-defined".
Here, if x = +2 or if x = -2, then the function would be asked to divide by zero.
Therefore, these two values will NOT be in the domain.

2) You also have a square root. In real numbers, you cannot have the square root of a negative number. In this function, the value x^2 - 4 will be negative when x is between -2 and + 2. Therefore this whole interval (-2, +2) is NOT in the domain.

There are no other problems, so the rest of the numbers are OK

The domain is, therefore, whatever is left:

Domain = (-∞, -2) U (+2, +∞)

The round brackets mean that the border values (for example -2) is NOT part of the domain. If it had been included in the interval, we would have used a square bracket.

∞ is the symbol for infinity (it always gets a round bracket because it is not a "well-defined" number)
I used the + sign for clarity, but most people would not bother (in this context, it is not needed)
(-∞, -2) U (2, ∞) means exactly the same thing as (-∞, -2) U (+2, +∞)

U is the "union" sign (also called "or") which means that the answer can come from either portion of the interval.

Here is another way of writing the same interval. The letter ℝ means "all the real numbers" and here we show the domain as being what is left after we remove the problem interval.