Question

The domain of the function f(x)=2x+3/underroot (x-2)(3-x) is​

Answer 1

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

Given function is

$\rm :\longmapsto\:f(x) = \dfrac{2x + 3}{ \sqrt{(x - 2)(3 - x)} }$

$\rm :\longmapsto\:So \: f(x) \: is \: defined \: if$

$\rm :\longmapsto\:(x - 2)(3 - x) > 0$

$\rm :\longmapsto\: - (x - 2)(x - 3) > 0$

$\rm :\longmapsto\:(x - 2)(x - 3) < 0$

$\bf\implies \:2 < x < 3$

$\bf\implies \: x \: \in \: (2, \: 3)$

Additional Information :-

Let us assume that a > b then

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

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

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

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

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).