Find the domain and the range of each of the following real

Given: $f(x)=\frac{3 x-2}{x+2}$

Need to find: Where the functions are defined.

Let, $f(x)=\frac{3 x-2}{x+2}=y$ .......(1)

To find the domain of the function f(x) we need to equate the denominator of the function to 0.

Therefore

$x+2=0$

$\Rightarrow x=-2$

It means that the denominator is zero when $x=-2$

So, the domain of the function is the set of all the real numbers except $-2$.

The domain of the function, $\operatorname{Df}(x)=(-\infty,-2) \cup(-2, \infty)$.

Now, to find the range of the function we need to interchange $x$ and $y$ in the equation no. (1)

So the equation becomes,

$\frac{3 y-2}{2+y}=x$

$\Rightarrow 3 y-2=2 x+x y$

$\Rightarrow 3 y-x y=2 x+2$

$\Rightarrow y=\frac{2 x+2}{3-x}=f\left(x_{1}\right)$

To find the range of the function $f\left(x_{1}\right)$ we need to equate the denominator of the function to 0 .

Therefore

$3-x=0$

$\Rightarrow x=3$

It means that the denominator is zero when $x=3$

So, the range of the function is the set of all the real numbers except 3 .

The range of the function, $\mathrm{Rf}(\mathrm{x})=(-\infty, 3) \cup(3, \infty)$.