Question

1. find the inverse of the function.

$g(x) = \frac{2x - 8}{4x + 5}$

1. write your answer in the form $\frac{ax + b}{cx + d}$, where a, b, c, and d are integers.

$g^{-1}(x) = \square$

Explanation:

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

$y = \frac{2x - 8}{4x + 5}$

Step2: Swap $x$ and $y$

$x = \frac{2y - 8}{4y + 5}$

Step3: Cross-multiply to eliminate fraction

$x(4y + 5) = 2y - 8$
$4xy + 5x = 2y - 8$

Step4: Isolate terms with $y$

$4xy - 2y = -8 - 5x$

Step5: Factor out $y$

$y(4x - 2) = -5x - 8$

Step6: Solve for $y$

$y = \frac{-5x - 8}{4x - 2}$
We can factor out $-1$ from numerator and denominator to simplify:
$y = \frac{-(5x + 8)}{2(2x - 1)} = \frac{5x + 8}{-4x + 2}$ (either form is valid, but we present the one with integer coefficients as required)

Answer:

$\boldsymbol{\frac{-5x - 8}{4x - 2}}$ (or equivalent $\boldsymbol{\frac{5x + 8}{-4x + 2}}$)