The domain of a function is all values of x the function can assume.
In this function, x can assume any real value but zero (because there would be a fraction with zero in the denominator), so the domain is:
[tex]D=\mleft\lbrace x\in\R\mright|x\ne0\}[/tex]The range of a function is all values of f(x) the function can assume.
In this function, f(x) can assume any positive number (greater than zero), so the range is:
[tex]R=\mleft\lbrace f(x\mright)\in\R|f(x)>0\}[/tex]In order to find the inverse function, we just need to switch x by f^-1(x) and f(x) by x, and then isolate f^-1(x), so we have:
[tex]\begin{gathered} f(x)=\frac{16}{x^4} \\ x=\frac{16}{(f^{-1}(x))^4} \\ (f^{-1}(x))^4=\frac{16}{x} \\ f^{-1}(x)=\sqrt[4]{\frac{16}{x}} \end{gathered}[/tex]f(x) is a function, since any value of x has only one corresponding value of f(x).
