Question

Determine the Domain and Range for the following functions:
f(x)=√x∧2-169

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm :\longmapsto\:y = \sqrt{ {x}^{2} - 169 }$

Domain

$\rm :\longmapsto\: \sqrt{ {x}^{2} - 169 } \: is \: defined \: if \: {x}^{2} - 169 \geqslant 0$

$\rm :\longmapsto\: {x}^{2} - {13}^{2} \geqslant 0$

$\rm :\longmapsto\:(x + 13)(x - 13) \geqslant 0$

$\rm :\implies\:x \leqslant - 13 \: \: or \: \: x \geqslant 13$

$\bf\implies \:x \: \in \: ( - \infty , - 13] \: \cup \: [13, \: \infty )$

Range

$\rm :\longmapsto\:y = \sqrt{ {x}^{2} - 169}$

$\rm :\longmapsto\: {y}^{2} = {x}^{2} - 169$

$\rm :\longmapsto\: {x}^{2} = 169 + {y}^{2}$

$\rm :\longmapsto\:x = \sqrt{169 + {y}^{2} }$

$\bf\implies \:y \geqslant 0$

$\bf\implies \:y \: \in \: [0, \: \infty )$

Basic Concept Used :-

Domain :- Let f(x) be a function, then set of those values of x where f(x) is well defined is called domain.

Range :-

To find the range of f(x)

Step : - 1. Let y = f(x)

Step :- 2. Express x in terms of y, say x = g(y).

Step :- 3. Find the domain of g(y).

Step :- 4. This will be the range of f(x).

Additional Information :-

Let assume that a > b then

$\rm :\longmapsto\:(x - a)(x - b) < 0 \implies \: a < x < b$

$\rm :\longmapsto\:(x - a)(x - b) \leqslant 0 \implies \: a \leqslant x \leqslant b$

$\rm :\longmapsto\:(x - a)(x - b) > 0 \implies \: x < b \: \: or \: \: x > a$

$\rm :\longmapsto\:(x - a)(x - b) \geqslant 0 \implies \: x \leqslant b \: \: or \: \: x \geqslant a$