show that f is one to one on its domain: f(x)=x3-1

Question

Consider the function

Answer 1

Let y=f(x)=x³-1,

x³=y+1,

x=∛(y+1). 

If g(y)=x=∛(y+1), then g(x)=∛(x+1).

f(g(x))=g(f(x))=x so g(x)=f^-1(x) (the inverse of f).

y=f(x): For every x there is a unique y;

y=g(x): For every y there is a unique x.

So the mapping is one-to-one.

so f^-1(x)=∛(x+1).