Question

Find the range of each of the following functions:

(i) $f\left(x\right)=2-3x,xϵR,x>0$

(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) Here f(x) = 2 - 3x

Since $xϵR$ and x > 0

$\therefore 3x>0\Rightarrow -3x<0\Rightarrow 2-3x<2$

$\therefore$ Range of function $=(\infty ,2)$

(ii) Here $f\left(x\right)=x^2+2$

Since $xϵR$

$\therefore x^2\ge 0$ for all $xϵR$

$\Rightarrow x^2+2\ge 2$

$\therefore$ Range of function

$=\{aϵR,a\ge 2\forall aϵR\}$

$=[2,\infty )$

(iii) Here $f\left(x\right)=x$

Since $xϵR$

$\therefore$ Range of function = R.