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.