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

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

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Answer 1 · 2016-03-07T22:39:28

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

Explanation:

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

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

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

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

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

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

We see that $y \neq \frac{3}{5}$ $y != 3/5$ .

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

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