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Question
Find the domain and range of the following function.
f(x) = `sqrt((x - 3)/(7 - x))`
Solution
f(x) = `sqrt((x - 3)/(7 - x))`
f(x) is defined, if `(x - 3)/(7 - x)` ≥ 0 and x ≠ 7
`(x - 3)/(7 - x)` ≥ 0, if x – 3 ≥ 0 and 7 – x > 0
or x – 3 ≤ 0 and 7 – x < 0
If x – 3 ≤ 0 and 7 – x < 0, then
x ≤ 3 and 7 < x
i.e., x ≤ 3 and x > 7
This is not possible.
∴ `(x - 3)/(7 - x)` ≥ 0, if x – 3 ≥ 0 and 7 – x > 0
i.e., if x ≥ 3 and 7 > x
i.e., if 3 ≤ x < 7
∴ Domian = {x/x ∈ R, 3 ≤ x < 7}
= [3, 7)
Let y = `sqrt((x - 3)/(7 - x))` ≥ 0 for all x ∈ [3, 7)
∴ Range = `[0, ∞)`
∴ Domian = [3, 7), Range = `[0, ∞)`
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