How do you find the inverse of y=(3x-7)/(x+9)?

Apr 17, 2018

${f}^{-} 1 \left(x\right) = \frac{9 x + 7}{3 - x}$

Explanation:

let, $y = f \left(x\right) = \frac{3 x - 7}{x + 9}$ then we have
$x = {f}^{-} 1 \left(y\right)$
$\Rightarrow y = \frac{3 x - 7}{x + 9} \Rightarrow y \left(x + 9\right) = 3 x - 7 \Rightarrow x y + 9 y = 3 x - 7$
$\Rightarrow 9 y + 7 = x \left(3 - y\right) \Rightarrow x = \frac{9 y + 7}{3 - y}$
since $x = {f}^{-} 1 \left(y\right) \Rightarrow {f}^{-} 1 \left(y\right) = \frac{9 y + 7}{3 - y} \Rightarrow {f}^{-} 1 \left(x\right) = \frac{9 x + 7}{3 - x}$

Apr 17, 2018

$f {\left(x\right)}^{-} 1 = \frac{- 9 x - 7}{3 + x}$

Explanation:

The inverse of a function switches the imput value and the output value. One easy way to solve inverse functions is by simply switching where the $x ' s \mathmr{and} y ' s a r e$. So...
$f \left(x\right) = \frac{3 x - 7}{x + 9}$ turns into $x = \frac{3 y - 7}{y + 9}$
Then from here on it is basic algebra.
$x = \frac{3 y - 7}{y + 9}$
$x \cdot \left(y + 9\right) = \left(3 y - 7\right)$
$x y + 9 x = 3 y - 7$
$3 y + x y = - 9 x - 7$
$y \left(3 + x\right) = - 9 x - 7$
$f {\left(x\right)}^{-} 1 = \frac{- 9 x - 7}{3 + x}$

If you need any more of an explanation, I will add them in