What is the range the function $h(x)=\frac{x-2}{x+3}$ ?
What is the range the function $h(x)=\frac{x-2}{x+3}$ ?
$(-\infty, 2) \cup(2, \infty)$
$(-\infty, 1) \cup(1, \infty)$
$(-\infty,-3) \cup(-3, \infty)$
$(-\infty,-1) \cup(-1, \infty)$
Solution
To find, range of the function $h(x)=\frac{x-2}{x+3}$
Let $h(x)=y$
i.e. set of possible values of $y$ will give the range of $h(x)$.
$
\begin{aligned}
& \Rightarrow & \frac{x-2}{x+3} & =y \\
& & x-2 & =x y+3 y \\
\Rightarrow & & x-x y & =3 y+2 \\
\Rightarrow & & x & =\frac{3 y+2}{1-y}
\end{aligned}
$
which is not defined for $y=1$
Hence, range $=R-\{1\}$ or $(-\infty,-1) \cup(1, \infty)$