Identifying domains

Question

Consider $f(x)=\sqrt{4-e^{4x}}$

a) What is the domain?

b) Find a formula for the inverse?

c) What is the domain of the inverse?

Struggling to fully comprehend how the domains for these functions are found. for part one i just solved for the function being equal or greater than zero and got $x \leq \frac{1}{4}\ln(4)$.

My inverse function was $f^{-1}(x)$ = $\frac{1}{4}\ln(4-x^2)$

The answer for part c is $[0,2)$.Can someone please explain step by step how that can be found.

Answer 1

To have $\ln(4-x^2)$ defined, we need $4-x^2>0$

$\iff x^2<4\iff-2<x<2$

Answer 2

I think your function is from $\mathbb{R}$ to $\mathbb{R}.$ Then $$4-e^{4x}\ge 0$$ $$2\ge e^{2x}\ge 0$$ $$x\in(-\infty,\dfrac{\ln 2}{2}]$$ is the domain of this function. Your inverse is correct and note that $\ln x$ is defined for all $x>0$ which is very smiler to the square root function.