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

Hint: The domain of a function is the set of all acceptable input values (X-values). The range of a function is the set of all output values hence domain is given for (i) the domain is given that is $x > 0$ , (ii) $x{\text{ is a real number }}$and for (iii) the domain of the function is Real Number put the domain what we get as output that is the range of the function.

Complete step-by-step answer:
In this question we have to find the range of the function
Range is the output value of the function if we put the domain as the input . Generally if we draw the graph of function the Y - axis represent the Range of the function and X- axis represent the domain of the function .
So in the First part ${\text{F(}}x{\text{) = 2 - 3}}x,x \in {\text{R,}}x > 0$ the domain is given that is $x > 0$ or $x$ is a positive real number
So from $x > 0$ we have to find the range of ${\text{F(}}x{\text{) }}$
$x > 0$
Now multiple by $ - 3$ in whole equation ,
$ - 3x < 0$ as we multiple by negative the inequality sign will change ,
Now Add $2$ on both side of equation ,
$2 - 3x < 2$
As we know that the ${\text{F(}}x{\text{) = 2 - 3}}x$
${\text{F}}(x) < 2$
Hence range of ${\text{F(}}x{\text{) = 2 - 3}}x,x \in {\text{R,}}x > 0$ is $\left( { - \infty ,2} \right)$
For part (ii) ${\text{F(}}x{\text{) = }}{x^2} + 2,x{\text{ is real number }}$
As we know that the value of ${x^2}$ is always positive mean that
${x^2} > 0$
Now add $2$ on both side
${x^2} + 2 > 2$
${\text{F}}(x) > 2$
Hence the range of ${\text{F(}}x{\text{) = }}{x^2} + 2,x{\text{ is real number }}$ is $\left( {2,\infty } \right)$ .
For the part (iii) ${\text{F(}}x{\text{) = }}x,x{\text{ is a real number }}$
In this function whatever we put in it as input we get the output the same value ,
So it is given that the domain of the function is Real Number hence the range is also the real number
The range of ${\text{F(}}x{\text{) = }}{x^2} + 2,x{\text{ is real number }}$ is $\left( { - \infty ,\infty } \right)$

Note: In this question we also find the domain of the function be drawing the graph of the given function e.g ${\text{F(}}x{\text{) = 2 - 3}}x,x \in {\text{R,}}x > 0$ if we draw graph as $y = 2 - 3x$ it is equation of line draw it after that the range of x-axis give the domain while the range of y-axis give the Range of the function .