Question

Find the inverse and the domain and range and state wether it's a functionf(x) =16/x^4

Answer 1

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:

`D=\mleft\lbrace x\in\R\mright|x\\e0\}`

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:

`R=\mleft\lbrace f(x\mright)\in\R|f(x)>0\}`

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:

`\begin{gathered} f(x)=(16)/(x^4) \\ x=(16)/((f^(-1)(x))^4) \\ (f^(-1)(x))^4=(16)/(x) \\ f^(-1)(x)=\sqrt[4]{(16)/(x)} \end{gathered}`

f(x) is a function, since any value of x has only one corresponding value of f(x).