Question

Find the domain and range of the following function.

f(x) = `sqrt((x - 3)/(7 - x))`

Answer 1

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, ∞)`