Question

algebra
review passage
find the inverse of the given function if it exists.
question
$f(x) = \frac{8}{5x + 7}$
select a single answer
$f^{-1}(x) = \frac{8 - 7x}{5x}$
$f^{-1}(x) = \frac{5x + 7}{8}$
the function does not have an inverse.
$f^{-1}(x) = \frac{7x - 8}{5x}$

Explanation:

Step1: Replace $f(x)$ with $y$

$y = \frac{8}{5x+7}$

Step2: Swap $x$ and $y$

$x = \frac{8}{5y+7}$

Step3: Solve for $y$

First, cross-multiply: $x(5y+7) = 8$
Expand: $5xy + 7x = 8$
Isolate terms with $y$: $5xy = 8 - 7x$
Solve for $y$: $y = \frac{8-7x}{5x}$

Step4: Replace $y$ with $f^{-1}(x)$

$f^{-1}(x) = \frac{8-7x}{5x}$

Answer:

$f^{-1}(x) = \frac{8-7x}{5x}$