Question

Draw the graph of f(x) = x 2 − 3 and find its domain and range.

Answer 1

Answer:

Given,

$\sf \: f(x) = {x}^{2} - 3 \\ \sf \:$

The function f (x) is defined for all real values of x

So,

$\sf \: domain \: \: D_f = \mathbb{R} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = ( - \infty \:, \infty)$

To find range, we assumed that,

$\sf \: y = f(x) = {x}^{2} - 3 \\ \sf \implies \: y = {x}^{2} - 3 \\ \sf \implies \: {x}^{2} = y + 3 \\ \sf \implies \: x = \sqrt{y + 3} \\ \bf \: now \: \: x = \sqrt{y + 3} \: \: will \: exist \: iff \\ \\ \sf \: \: \: \: \: \: y + 3 \geqslant 0 \\ \sf \implies \: y \geqslant - 3 \\ \\ \sf \: \: \green {\boxed{ so \: range \: R_f = [3, \infty)}}$