Question

What is the domain and range of the functions below;
A). y=2x²-3
B). f(x)= -3X²+1

Answer 1

$\large\underline{\sf{Solution-(a)}}$

Given function is

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

Domain

$\rm :\longmapsto\:Since \: {2x}^{2} - 3 \: is \: a \: polynomial \: function$

$\rm :\implies\: {2x}^{2} - 3 \: is \: defined \: for \: all \: real \: values \: of \: x.$

$\bf\implies \:Domain \: of \: y \: = \: x \: \in \: R$

Range

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

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

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

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

$\rm :\longmapsto\:x \: is \: defined \: if \: \dfrac{y + 3}{2} \geqslant 0$

$\rm :\longmapsto\:y + 3 \geqslant 0$

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

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

$\large\underline{\sf{Solution-(b)}}$

Given function is

$\rm :\longmapsto\:f(x) = y = - {3x}^{2} + 1$

Domain

$\rm :\longmapsto\:Since \: { - 3x}^{2} + 1\: is \: a \: polynomial \: function$

$\rm :\implies\: { - 3x}^{2} + 1 \: is \: defined \: for \: all \: real \: values \: of \: x.$

$\bf\implies \:Domain \: of \: y \: = \: x \: \in \: R$

Range

$\rm :\longmapsto\: y = - {3x}^{2} + 1$

$\rm :\longmapsto\: {3x}^{2} = 1 - y$

$\rm :\longmapsto\: {x}^{2} = \dfrac{1 - y}{3}$

$\rm :\longmapsto\:x = \sqrt{\dfrac{1 - y}{3} }$

$\rm :\longmapsto\:x \: is \: defined \: if \: \dfrac{1 - y}{3} \geqslant 0$

$\rm :\longmapsto\:1 - y \geqslant 0$

$\rm :\longmapsto\: - y \geqslant - 1$

$\rm :\longmapsto\:y \leqslant 1$

$\bf\implies \:y \: \in \: ( - \infty ,1]$

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).