Question

If f (x) = 2x2 – 5for - ♾ < x < 0,what is f-?(-2)?

Answer 1

To find the inverse, we:

• change f(x) to y

,

• interchange x and y

,

• solve for y

Thus,

$\begin{gathered} f(x)=2x^2-5 \\ y=2x^2-5 \\ x=2y^2-5 \\ 2y^2=x+5 \\ y^2=\frac{x+5}{2} \\ y=\pm\sqrt[]{\frac{x+5}{2}} \\ f^{-1}(x)=-\sqrt[]{\frac{x+5}{2}} \end{gathered}$

We take the "negative" part of the function since it is defined for - ♾ < x < 0.

We found inverse of f.

Now, to find f^(-1) (-2), we put -2 into the inverse and evaluate.

$\begin{gathered} f^{-1}(x)=-\sqrt[]{\frac{x+5}{2}} \\ f^{-1}(-2)=-\sqrt[]{\frac{-2+5}{2}} \\ =-\sqrt[]{\frac{3}{2}} \end{gathered}$