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Question
Find the domain and range of the following function.
f(x) = `sqrt((x - 2)(5 - x)`
Solution
f(x) = `sqrt((x - 2)(5 - x)`
For f to be defined,
(x – 2) (5 – x) ≥ 0
∴(x – 2) (x – 5) ≤ 0
∴ 2 ≤ x ≤ 5 ...`[("The solution of" (x - "a") (x - "b") ≤ 0),("is" "a" ≤ x ≤ "b""," "for" "a" < "b")]`
Domain = [2, 5]
(x – 2) (5 – x) = – x2 + 7x – 10
= `-(x - 7/2)^2 + 49/4 - 10`
= `9/4 - (x - 7/2)^2 ≤ 9/4`
∴ `sqrt((x - 2)(5 - x)) ≤ sqrt(9/4) ≤ 3/2`
∴ Range of f = `[0, 3/2]`
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