Question

Find the range of each of the following functions
(i) $f\left(x\right)=2-3x,$ $x\in R,x>0$
(ii) $f\left(x\right)=$ $x^2+2$, $x$ is a real number
(iii) $f\left(x\right)=x,x$ is a real number

Answer 1

(i) Given $x>0$
$\Rightarrow 3x>0$

$\Rightarrow -3x<0$
$\Rightarrow 2-3x<2$
$\Rightarrow f\left(x\right)<2$
$\therefore$ Range of $f=(-\infty ,2)$
(ii) Since, for any real number $x$, $x^2\ge 0$
$\Rightarrow x^2+2\ge 0+2$
$\Rightarrow x^2+2\ge 2$
$\Rightarrow f\left(x\right)\ge 2$
$\therefore$ Range of $f=[2,\infty )$
(iii) $f\left(x\right)=x,x$ is a real number
It is clear that the range of $f$ is the set of all real numbers
$\therefore$ Range of $f=R$