Answer:
[tex]f^-^1(x)=\frac{(x+1)^2}{16}[/tex]
Step-by-step explanation:
We have the function: [tex]f(x)=-4\sqrt{x} -1[/tex] and we have to find the inverse.
Observation: [tex]y=f(x)[/tex], then [tex]y=-4\sqrt{x} -1[/tex]
You can obtain the inverse of a function by switching the x and y values. This means:
[tex]y=-4\sqrt{x} -1\\x=-4\sqrt{y'} -1[/tex]
We're going to call [tex]f^-^1(x)=y'[/tex]. Now we have to clear "y'".
First we have to add (1) in both sides of the equation:
[tex]x=-4\sqrt{y'} -1\\x+1=-4\sqrt{y'}-1+1\\x+1=-4\sqrt{y'}[/tex]
Now divide in (-4) both sides.
[tex]x+1=-4\sqrt{y'}\\\frac{(x+1)}{(-4)} =\frac{(-4)\sqrt{y'}}{(-4)} \\\\\frac{(x+1)}{(-4)} =\sqrt{y'}[/tex]
And for our last step we have to square both sides:
[tex]\frac{(x+1)}{(-4)} =\sqrt{y'}\\\\(\frac{(x+1)}{(-4)})^2=(\sqrt{y'})^2\\\frac{(x+1)^2}{(-4)^2}=y'\\\\\frac{(x+1)^2}{16}=y'[/tex]
Then the third option is the correct: [tex]f^-^1(x)=\frac{(x+1)^2}{16}[/tex]
