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

To determine the domain of the inverse function of ( f(x) = \frac{-12x - 7}{x + 3} ), we first need to understand the function itself, particularly any restrictions on its domain. The function is a rational function, and the only restriction occurs when the denominator equals zero. Setting the denominator equal to zero, we find that:

[ x + 3 = 0 \implies x = -3. ]

Thus, the function is not defined at ( x = -3 ). This tells us that the domain of ( f(x) ) includes all real numbers except for ( x = -3 ).

When dealing with inverse functions, the roles of the input and output are switched. Therefore, the domain of the inverse function corresponds to the range of the original function. Since ( f(x) ) is continuous and only has one discontinuity at ( x = -3 ), we can find the value of ( f(-3) ) to identify another point of note. However, we can observe that the function ( f(x) ) approaches a vertical asymptote at ( x = -3 ) but maintains continuity elsewhere.

To find the range, we