Find the domain of each of the following real valued functions of real variable:
We know the square of a real number is never negative.
Clearly, f(x) takes real values only when x – 2 and 3 – x are both positive or negative.
(a) Both x – 2 and 3 – x are positive
x – 2 ≥ 0 ⇒ x ≥ 2
3 – x ≥ 0 ⇒ x ≤ 3
Hence, x ≥ 2 and x ≤ 3
∴ x ∈ [2, 3]
(b) Both x – 2 and 3 – x are negative
x – 2 ≤ 0 ⇒ x ≤ 2
3 – x ≤ 0 ⇒ x ≥ 3
Hence, x ≤ 2 and x ≥ 3
However, the intersection of these sets in null set. Thus, this case is not possible.
In addition, f(x) is also undefined when 3 – x = 0 because the denominator will be zero and the result will be indeterminate.
3 – x = 0 ⇒ x = 3
Hence, x ∈ [2, 3] – {3}
∴ x ∈ [2, 3)
Thus, domain of f = [2, 3)