fhow to find the domain and range of root of x2-4 - Maths - Relations and Functions

Question

fhow to find the domain and range of root of x2
-4

Shridhar Kaushik · Accepted Answer

We have,

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

We observe that f(x) is defined for all x satisfying

$x^{2} - 4 \geq 0 \Rightarrow (x + 2) (x - 2) \geq 0 \Rightarrow x \geq 2 or x \leq - 2 Hence the domain is (- \infty, - 2] \cup [2, \infty) . Let y = f (x) . Then, y = \sqrt{x^{2} - 4} \Rightarrow y^{2} = x^{2} - 4 \Rightarrow x^{2} = y^{2} + 4 \Rightarrow x = \sqrt{y^{2} + 4}$

Clearly, x will take real values if

$y^{2} + 4 \geq 0$

Which is true for all real values of y.

Also, $y = \sqrt{x^{2} - 4} \geq 0$ for all $x \in (- \infty, - 2] \cup [2, \infty)$ .

$∴ y \in [0, \infty) . Hence, Range (f) = [0, \infty) .$

fhow to find the domain and range of root of x2 -4

fhow to find the domain and range of root of x²-4