For a function f defined by f : x → 4x² - 3, find a domian of x corresponding to range 1 < f(x) < 13.

f(x) = 4x^2 - 3.                                                                                          sqrt(3)
Set this equal to 1 first, and solve for x:  4x^2 = 3, and x^2 =3/4, so
                                                                    sqrt(3)
                                                              x = -----------
                                                                          2                                           

This is approx. 0.866.  It's the lower end of the domain.

Set f(x) = 13 to find the higher end of the domain:  4x^2 - 3 = 13
Simplifying, 4x^2 = 16, x^2 = 4, and x=2

The domain here does not include the end points sqrt(3) / 2 or 2; it is

( sqrt(3) / 2, 2)   (answer)