​ Find the domain and range of the following real functions f(x) = √x^2+4 ​

Question

Find the domain and range of the following real functions

f(x) = \(\sqrt{x^2+4}\)

Answer 1

Given, f(x) =\(\sqrt{x^2+4}\)

Domain of f: We observe that f(x) is defined for all x satisfying x² + 4 ≥ 0 

∴ Domain = The set of all real number = R 

∴ Range of: 

Let y = f (x) 

y = \(\sqrt{x^2+4}\)

y = \(\sqrt{y^2-4}\) 

x is defined, 

y² – 4 ≥ 0 (y + 2)(y − 2) ≥ 0 

∴ Range of f = ( – ∞, – 2] ∪ [2, ∞)