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

##### 1 Answer

#### Explanation:

Let

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

Then we are constrained to

For any

Note that

For small positive values of

For large positive values of

Since the function is also continuous, the range is

So for any value of

This defines our inverse function:

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

That is

We have shown (informally) that this exists, but there is no algebraic solution for

The graph of

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 }#