Question

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

Answer 1

$\left(i\right)$ Given: $f\left(x\right)=2-3x,x\in R,x>0$
Range of function
As $x>0$
$\Rightarrow 3x>0$
Multiply both side by $-1$, the inequality sign changes
$\Rightarrow -3x<0$
$\Rightarrow 2-3x$ is less than $2$ [adding $2$ both sides)
Therefore, the value of $2-3x$ is less than 2
Hence, Range$=(-\infty ,2)$
Range of the given function is $(-\infty ,2)$

$\left(ii\right)$ Given: $f\left(x\right)=x^2+2,x$ is a real number
Range of function:
As we know,
$\Rightarrow x^2\ge 0$
$\Rightarrow x^2+2\ge 2$ [adding $2$ both sides]
Therefore,the value of given function is always greater than or equal to $2$
Hence, Range of the given function is $[2,\infty )$

$\left(iii\right)$ Given:$f\left(x\right)=x,x$ is a real number
Range of function $f\left(x\right)=x\in R$
So, $f\left(x\right)\in R$
$\Rightarrow y\in R$
Therefore, the range of the given function is $R$ or $(-\infty ,\infty )$.