# What is the domain and range of 3x-2/5x+1 and the domain and range of inverse of the function?

## What is the domain and range of 3x-2/5x+1 and the domain and range of inverse of the function?

Mar 7, 2016

Domain is all reals except $- \frac{1}{5}$ which is the range of the inverse.
Range is all reals except $\frac{3}{5}$ which is the domain of the inverse.

#### Explanation:

$f \left(x\right) = \frac{3 x - 2}{5 x + 1}$ is defined and real values for all $x$ except $- \frac{1}{5}$, so that is the domain of $f$ and the range of ${f}^{-} 1$

Setting $y = \frac{3 x - 2}{5 x + 1}$ and solving for $x$ yields

$5 x y + y = 3 x - 2$, so

$5 x y - 3 x = - y - 2$, and therefore

$\left(5 y - 3\right) x = - y - 2$, so, finally

$x = \frac{- y - 2}{5 y - 3}$.

We see that $y \ne \frac{3}{5}$.

So the range of $f$ is all reals except $\frac{3}{5}$. This is also the domain of ${f}^{-} 1$.