The function $f$ is defined by $f(x) = e^{-x^4}, x \in \mathbb{R}$ - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 1

To find the inverse of the function, we start with the equation:

$y = e^{-x^4}$

Taking the natural logarithm on both sides gives:

$\ln(y) = -x^4$

Rearranging this equation leads to:

$-\ln(y) = x^4$

And therefore:

$x = \sqrt[4]{-\ln(y)}$

Thus, the inverse function can be expressed as:

$$f^{-1}(x) = 4 + \ln(x), \text{ for } x > 0.$