What is the inverse of the function f(x) = (-12x - 7)/(x + 3)?

To determine the inverse of the function f(x) = ((-12x - 7)/(x + 3)), we start by letting y be equal to f(x):

[ y = \frac{-12x - 7}{x + 3} ]

Next, to find the inverse, we switch the roles of y and x, and then solve for y:

[ x = \frac{-12y - 7}{y + 3} ]

Multiplying both sides by (y + 3) to eliminate the fraction gives:

[ x(y + 3) = -12y - 7 ]

Expanding the left side results in:

[ xy + 3x = -12y - 7 ]

Reorganizing the equation to isolate terms containing y on one side yields:

[ xy + 12y = -7 - 3x ]

Factoring y out from the left side results in:

[ y(x + 12) = -7 - 3x ]

Finally, dividing both sides by the expression (x + 12) provides the expression for y:

[ y = \frac{-7 - 3x}{x + 12} ]

This simplifies