Question

f is a 1-1 function. If f(x) is increasing, is f(inverse)(x) increasing. why

Answer 1

Let f be a 1-1 function and f be an increasing function


So because f is an increasing function, this means that if a < b, then f(a) < f(b)


We can flip that to say that if f(a) < f(b) then a < b


Now apply the inverse to both sides to get


f^(-1){f(a)} < f^(-1){f(b)}


a < b


So this adds further support that the function is increasing.


This also shows us that the inverse function is also an increasing function because we can rewrite that first line as f^(-1)( p ) < f^(-1)( q ) ===> p < q, where p = f(a) and q = f(b)

Answer 2

thx man this helps