Question

Find the domain and range of the real valued function:

(iii)  \[f\left( x \right) = \sqrt{x - 1}\]

 

Answer 1

Given:

\[f\left( x \right) = \sqrt{x - 1}\]

Domain ( f ) : Clearly, f (x) assumes real values if x - 1 ≥ 0 ⇒ x ≥ 1 ⇒ x ∈ [1, ∞) .

Hence, domain (f) = [1, ∞)
Range of f : For x ≥  1, we have:
x - 1 ≥ 0

\[\Rightarrow \sqrt{x - 1} \geq 0\]

⇒ f (x) ≥ 0
Thus, f (x) takes all real values greater than zero.
Hence, range (f) = [0, ∞) .