Question

Find the range of each of the following functions:

(i) $f\left(x\right)=2-3x,xϵR$ and $x>0$

(ii) $g\left(x\right)=x^2+2,xϵR$

Answer 1

We have, $f\left(x\right)=2-3x$, where $xϵR$ and $x>0$

Now, $x>0\Rightarrow 3x>0\Rightarrow -3x<0$

$\Rightarrow -3x+2<0+2\Rightarrow 2-3x<2$

$\Rightarrow f\left(x\right)<2\Rightarrow f\left(x\right)ϵ(-\infty ,2)$

Hence, range (f) $=(-\infty ,2)$

(ii) We have, $g\left(x\right)=x^2+2,xϵR$

Now, $xϵR\Rightarrow x^2\ge 0\Rightarrow x^2+2\ge 0+2$

$\Rightarrow x^2+2\ge 2\Rightarrow g\left(x\right)\ge 2$

$\Rightarrow g\left(x\right)ϵ[2,\infty )$

Hence, range (g) $=[2,\infty )$.