Question

Please use the photo attached for the question and answers.

Answer 1

In order to find the inverse of f we need to take its equation and replace f(x) and x with x and f^-1(x). Then we have:

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

And we find the inverse function:

`\begin{gathered} x=-4\sqrt[]{f^(-1)(x)}-1 \\ x+1=-4\sqrt[]{f^(-1)(x)} \\ -((x+1))/(4)=\sqrt[]{f^(-1)\mleft(x\mright)} \\ f^(-1)(x)=((x+1)^2)/(16) \end{gathered}`

So this is the inverse function but we still have to find the inequality for if it has one. First is important to remember that the x in the last calculation replaced f(x). This means that the inequalities that f(x) meets are the same that x meets. So let's see, we have:

`f(x)=-4\sqrt[]{x}-1`

The square root of x can have as a result any number between 0 and infinite. This means that f(x) tends to negative infinite (when the square root tends to infinite) and that the maximum value of f(x) is:

`f(0)=-4\cdot0-1=-1`

This means that:

`f(x)\leq-1`

And since we replace f(x) by x then for the inverse function we have:

`f^(-1)(x)=((x+1)^2)/(16),x\leq-1`

Then the answer is the fourth option.