In this example, we are given f(x) = square root of (x^{2} - 9) and x is in the range of less than and equal to negative three. The question ask for f^{-1}(4).
How to find the value of the inverse function? The indenpendent variable of the function f is x and its range is x <= -3. The indenpendent
variable of the inverse function f^{-1} is y. So the question is ask for when y = 4, what is the corresponding value of x? We start from
y = square root of (x^{2} - 9). Substitute y = 4 into the equation. So 4 = square root of (x^{2} - 9). Now square booth side of the equation, we get
16 = x^{2} - 9. Simplify it, x^{2} = 16 + 9 = 25. Then x_{1} = 5 and x_{2} = -5. Since we are given x <= -3, so x_{1} = 5 is
not in the range of x, so x_{1} = 5 shoud be drop. So x = -5. That is, when y is 4, x is -5.