We need to find the inverse of y = 3x + 12. For this, express x in terms of y.

y = 3x + 12

=> 3x = y - 12

=> x = y/3 - 4

substitute x and y.

=> y = x/3 - 4

**The inverse of y = 3x + 12 is y = x/3 - 4**

The inverse of the function f(x) is f^-1(x).

To prove that f(x) has inverse, we'll have to prove first that f(x) is bijective.

To prove that f(x) is bijective, we'll have to prove that is one-to-one and on-to function.

1) One-to-one function.

We'll suppose that f(x1) = f(x2)

We'll substitute f(x1) and f(x2) by their expressions:

3x1 + 12= 3x2 + 12

We'll eliminate like terms:

3x1 = 3x2

We'll divide by 3:

x1 = x2

A function is one-to-one if and only if for x1 = x2 => f(x1) = f(x2).

2) On-to function:

For a real y, we'll have to prove that it exists a real x.

y = 3x + 12

We'll isolate x to the right side. For this reason, we'll subtract 12 both sides:

y - 12 = 3x

We'll use the symmetric property:

3x = y - 12

We'll divide by 3:

x = y/3 - 4

x is a real number => f(x) is an on-to function

Since the function is both, one to one and on-to function, then f(x) is bijective.

If f(x) is bijective => f(x) is invertible.

**The requested inverse function of f(x) = 3x + 12 is: f^-1(x) = x/3 - 4.**