Question

Find out if the function f(x) is invertible in (0,+inf). In affirmative case, find the inverse function.

Answer 1

what is the function

Answer 2

can you show the work, please

Answer 3

one way to show that it is invertible is to find the inverse and see if you have a function.
another way is to graph and see if it passes "horizontal line test" and a third way is to write
\[f(a)=f(b)\] and see if you can conclude that
\[a=b\]

Answer 4

this function says
1) cube
2) add 1
3) exponentiate

we can undo that by
1) take the log
2) subtract 1
3) take the cube root

so it is invertible and the inverse is
\[f^{-1}(x)=\sqrt[3]{\ln(x)-1}\]

Answer 5

if you want to "show work"
you can write
\[x=e^{y^3+1}\] and solve for y. you will get the same thing

Answer 6

Thanks