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:

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

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:

$f^{-1}(x)=\frac{(x+1)^2}{16},x\leq-1$

Then the answer is the fourth option.