Question

Find the domain and the range of the following function:
$f\left(x\right)=\frac{x-3}{2x+1}$

Answer 1

The function $f\left(x\right)=\frac{x-3}{2x+1}$ is defined only when $2x+1\ne 0x\ne \frac{-1}{2}$

So domain is $R-\left\{\frac{-1}{2}\right\}$

Let $y=\frac{x-3}{2x+1}(2x+1)y=x-32xy+y=x-3y+3=x(1-2y)x=\frac{y+3}{1-2y}$

This function is defined only when $1-2y\ne 0y\ne \frac{1}{2}$

So the range is $R-\left\{\frac{1}{2}\right\}$