Find the inverse function about a exponential related function

Question

Here is the function:$$y = 4x + {x^m},where{\text{ 0 < m}} \leqslant {\text{1;}}$$ Approximately results is acceptable.

Answer 1

If you follow André Nicolas's suggestion and use Newton iteration scheme, starting from $x_{old}$, the new predicate $x_{new}$ will be given by $$x_{new}=\frac{(m-1) x_{old}^m+y}{m x_{old}^{m-1}+4}$$ Update $x_{old}$ by $x_{new}$ and repeat.

Taking into account the condition ${\text{ 0 < m}} \leqslant {\text{1}}$, you could start iterating at $$x_{old}=\frac{y}{4} \left(1-\frac{m}{5}\right) $$ which gives the exact solution at the boundaries of $m$.

Just for illustration purposes, suppose that we choose $m=0.12345$ and $y=9876$. The previous formula gives an estimate equal to $2408.04$ and the first iterate will be $2468.34$ which is the solution. If we change $m=0.56789$, the first estimate will be $2188.58$ and the successive iterates will be $2447.96$ and $2447.99$ which is the solution.