Question

Look at the function and find parts A and B please

Answer 1

As given by the question

There are given that the function:

`f(x)=\sqrt[16]{x}`

Now,

To find the inverse, first interchange the function f(x) to y:

So,

`\begin{gathered} f(x)=\sqrt[16]{x} \\ y=\sqrt[16]{x} \end{gathered}`

Then,

Interchange the variable:

So,

`\begin{gathered} y=\sqrt[16]{x} \\ x=\sqrt[16]{y} \end{gathered}`

Then,

Solve the above equation for y

So,

`\begin{gathered} x=\sqrt[16y]{\square} \\ x=(y)^{(1)/(16)} \\ (x)^(16)=(y)^{(1)/(16)*16} \\ y=x^(16) \end{gathered}`

Then,

The inverse function is :

`f^(-1)(x)=x^(16)`

Hence, the correct option is (A) and the value is shown below:

`f^(-1)(x)=x^(16)`

(B):

For verify, put inverse function into the given function:

So,

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

And,

Put the value of f(x) into the inverse function:

So,

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

Hence proved.