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

Question

5310 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-03-07T22:39:28

Domain is all reals except $-1/5$ which is the range of the inverse.
Range is all reals except $3/5$ which is the domain of the inverse.

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

Setting $y=(3x-2)/(5x+1)$ and solving for $x$ yields

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

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

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

$x=(-y-2)/(5y-3)$ .

We see that $y != 3/5$ .

So the range of $f$ is all reals except $3/5$ . This is also the domain of $f^-1$ .