how to show algebraically f(x) = (x - 2)^3 + 8 is a one-to-one function?

Answer 1

it has an inverse, since

f^-1(x) = ∛(x-8) + 2

Also, you know that

g(x) = x^3

is 1-to-1, and f(x) is just g(x) shifted by (2,8).

More formally f(x) is one-to-one if

f(a) = f(b)
means that a = b

so, let's say that f(a) = f(b)
That means that
f(a) = f(b)
(a-2)^3+8 = (b-2)^3 + 8
(a-2)^3 = (b-2)^3
true iff a=b