You're right up to here for sure.

What I was taught after putting f(x) = y is make x the subject, written in terms of y:then switch x and y

so x = √(2y² + 12)

then rearrange for y

so i get

y = (1/2)√(x² - 12)

could anyone confirm this is right or wrong?

Finally replace x with and y with x:

Note that the plus/minus sign means it's not one to one and so not a function unless the domain is restricted