Given function: [tex]f(x)=-4\sqrt{x}-1[/tex].
We need to find the inverse of the given function f(x).
Let us write the function in terms of x and y first.
[tex]y=-4\sqrt{x}-1[/tex]
Now, in order to find the inverse, we need to switch x and y, we get
[tex]x=-4\sqrt{y}-1[/tex]
Now, we need to solve it for y.
In order to solve it for y, we need to isolate it for y.
Adding 1 on both sides, we get
[tex]x+1=-4\sqrt{y}-1+1[/tex]
[tex]x+1=-4\sqrt{y}[/tex]
Dividing both sides by -4, we get
[tex]\frac{x+1}{-4}=\sqrt{y}[/tex]
In order to get rid square root from right side, we need to square both side.
[tex](\frac{x+1}{-4})^2=(\sqrt{y})^2[/tex]
[tex]\frac{(x+1)^2}{16}=y[/tex]
Or [tex]y=\frac{1}{16}(x+1)^2[/tex]
Therefore, inverse function is
[tex]f^{-1}(x)=\frac{1}{16}(x+1)^2[/tex].
