Find the domain of real valued functions of real variable f (x) =$$\sqrt{\dfrac{{x-2}}{3-x}}$$

Asked by Aaryan | 1 year ago |  95

##### Solution :-

f (x) = $$\sqrt{\dfrac{{x-2}}{3-x}}$$

We know the square root of a real number is never negative.

f (x) takes real values only when x – 2 and 3 – x are both positive and 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 is null set. Thus, this case is not possible.

Hence, x ∈ [2, 3] – {3}

x ∈ [2, 3]

Domain (f) = [2, 3]

Answered by Aaryan | 1 year ago

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