Question

if a function f is defined as f: Z->Z, f(x)=x²-3 then find the range of the function​

Answer 1

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

Given that

$\rm :\longmapsto\:f : Z \: \to \: Z \: given \: by \: f(x) = {x}^{2} - 3$

Range of f(x)

$\rm :\longmapsto\:Let \: y \: = \: {x}^{2} - 3$

$\rm :\longmapsto\: {x}^{2} = y + 3$

$\rm :\longmapsto\:x = \sqrt{y + 3}$

$\rm :\implies\:x \: is \: defined \: if \: y + 3 \geqslant 0$

$\rm :\implies\:y \geqslant - 3$

$\bf\implies \:y \: \in \: [- 3, \: \infty )$

$\rm :\longmapsto\:As \: y \: \in \: Z$

$\bf\implies \:y \: = \: \{ - 3, - 2, - 1,0, 1,... \}$

Basic Concept Used :-

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).