What is the inverse of `y=3ln(5x)+x^3` ?

Let `f(x) = 3ln(5x)+x^3`

Let us assume that we are dealing with Real values and therefore the Real natural logarithm.

Then we are constrained to `x > 0` in order that `ln(5x)` be defined.

For any `x > 0` both terms are well defined and so `f(x)` is a well defined function with domain `(0, oo)`.

Note that `3ln(5)` and `x^3` are both strictly monotonic increasing on this domain so our function is too and is one-to-one.

For small positive values of `x`, the term `x^3` is small and positive and the term `3ln(5x)` is arbitrarily large and negative.

For large positive values of `x`, the term `3ln(5x)` is positive and the term `x^3` is arbitrarily large and positive.

Since the function is also continuous, the range is `(-oo, oo)`

So for any value of `y in (-oo, oo)` there is a unique value of `x in (0, oo)` such that `f(x) = y`.

This defines our inverse function:

`f^(-1)(y) = x : f(x) = y`

That is `f^(-1)(y)` is the value of `x` such that `f(x) = y`.

We have shown (informally) that this exists, but there is no algebraic solution for `x` in terms of `y`.

The graph of `f^(-1)(y)` is the graph of `f(x)` reflected in the line `y=x`.

In set notation:

`f = { (x, y) in (0, oo) xx RR : y = 3ln(5x)+x^3 }`

`f^(-1) = { (x, y) in RR xx (0, oo) : x = 3ln(5y)+y^3 }`