Q: Find the domain of the function:
d(x) = √(x-1)/(x-6)
A: To find the domain (which is what values x can be), think of what values x cannot be.
The two problem areas we have are (1) a square-root and (2) a denominator.
(1) Anything in under a square-root must be positive (greater than or equal to zero)… This is because we cannot take a square-root of a negative number.
So, (x – 1) ≥ 0
x – 1 ≥ 0
x ≥ 1
x must be greater than or equal to 1.
(2) The denominator can never equal zero. So,
(x – 6) ≠ 0
x ≠ 6
Now, we must put the two answers together: x must be greater than or equal to 1, but cannot be 6.
In interval notation, this is: [1, 6) U (6, ∞).