Answer:
[tex]f^{-1}(x)=4x^2-3[/tex]
Step-by-step explanation:
The Inverse of a Function
Given a function f(x), the inverse of f, called [tex]f^{-1}(x)[/tex] is a function that satisfies:
[tex]f(f^{-1}(x))=x[/tex]
The domain of f becomes the range of its inverse and vice-versa.
The procedure to find the inverse of the function is:
* Write the function as a two-variable equation:
* Solve the equation for x.
* Swap the variables
We are given the function:
[tex]\displaystyle f(x)=-\frac{1}{2}\sqrt{x+3}[/tex]
Defined in the interval x ≥ -3
Note f is always negative, thus its range is f(x) ≤ 0
Now we find the inverse. Call y=f(x):
[tex]\displaystyle y=-\frac{1}{2}\sqrt{x+3}[/tex]
Multiplying by 2:
[tex]\displaystyle 2y=-\sqrt{x+3}[/tex]
Squaring both sides:
[tex]\displaystyle 4y^2=x+3[/tex]
Subtracting 3:
[tex]\displaystyle 4y^2-3=x[/tex]
Swapping variables:
[tex]y=4x^2-3[/tex]
Thus:
[tex]\mathbf{f^{-1}(x)=4x^2-3}[/tex]
The domain of the inverse is the range of f, thus x is restricted to
x ≤ 0
