Question

Find the domain and range of the following function.

f(x) = `sqrt((x - 2)(5 - x)`

Answer 1

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]`