Expert: Steve Holleran - 9/29/2007

Question
This is one of the review problems of Pre-Calc: find the inverse of g: g(x)= x^2 + 2x + 3, when x is greater than or equal to -1. This is what I have so far:
y=x^2 + 2x + 3
x=y^2 + 2y + 3 (which can also be written as x-3=y(y+2) but i'm not sure that helps).

Answer
Hi Katie,

You've gotten a good start on this.  Take what you have, expand the right, and complete the square in y:

                x - 3 = y^2 + 2y

              x - 3 + 1    =  y^2 + 2y + 1

                   x - 2 = (y + 1)^2

            so    y + 1  = +/- sqrt(x - 2)

                     y = -1 +/- sqrt(x-2)

Now, you want y = -1 + sqrt(x-2) to be g inverse, because in the original function, the restriction x >= -1 means that you only considered the "right-hand" section of the parabola.

See, if you graph the original parabola, its vertex is at (-1,2) and it opens up.  Its not 1-1, so as it stands it has no inverse function.  By restricting x >=-1, that section of the parabola to the left of -1 is "erased", and the remaining right half is now a 1-1 function with an inverse.

I hope this helps you out.

Steve