How do I show that this function is one-to-one algebraically? f(x)= (x-2)^(3) +8

Question

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

`f(x)= (x-2)^(3) +8`

Answer 1

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.