What is the inverse of #y=3log(5x)+x^3#? ?

1 Answer
Oct 1, 2017

Answer:

#x = 3log(5y)+y^3#

Explanation:

Given:

#y = 3log(5x)+x^3#

Note that this is only defined as a real valued function for #x > 0#.

Then it is continuous and strictly monotonically increasing.

The graph looks like this:
graph{y = 3log(5x)+x^3 [-10, 10, -5, 5]}

Therefore it does have an inverse function, whose graph is formed by reflecting about the #y=x# line...
graph{x = 3log(5y)+y^3 [-10, 10, -5, 5]}

This function is expressible by taking our original equation and swapping #x# and #y# to get:

#x = 3log(5y)+y^3#

If this were a simpler function then we would typically want to get this into the form #y = ...#, but that is not possible with the given function using standard functions.