Inverse functions can be defined only for functions y = f(x) that have one to one mapping. For no two values of x in the domain of the function should the value of y be the same.
The function f(x) = 3x^2 + 1 has the following graph:
It can be clearly seen that there are two values of x that give the same value of y = f(x) for each value of y. An inverse of this function would have two possible values of f(x) for each value of x; the definition of functions does not allow this.
To be able to determine an inverse function, the domain of the function f(x) would have to be constrained so that there is a one to one mapping of x and f(x). This could be done by defining x as f(x) = 3x^2 + 1, x `>=` 0.
It has an inverse. But, the inverse isn't a function. To find the inverse:
1) make f(x) --> y
2) switch the x and y
3) solve for y now; get y by itself
Result is the inverse. So, for yours, step by step:
y = 3x^2 + 1
x = 3y^2 + 1 minus 1 from each side
x-1 = 3y^2 divide each side by 3
(x-1)/3 = y^2 take the square root of each side
(+-)sqrt[(x-1)/3)] = y
"+-" meaning take the positive and negative root of each side. Also, on this, if you notice, not all x's work. But, this is the inverse.