How do i show that this function is one-to-one algebraically?

For a one-to-one function, f(x) = f(y).
 So, lets find f(x) and f(y).

We know, 
f(x) = (x - 2)³ + 8

Now,
f(y) = (y - 2)³ + 8

Now, we said earlier, for a function to be one-to-one, f(x) = f(y).

Therefore,

            f(x) = f(y)
(x - 2)³ + 8 = (y - 2)³ + 8
      (x - 2)³ = (y - 2)³     
          x - 2 = y - 2
               x = y

Since we got x = y, we know for every x, there is one and only one y. Therefore, the function is one-to-one function.