"Given f(x)=-3x^2-7x+19 and g(x)=2x^2-4x-5 find the following:"

"e. g^-1(-2) ..... (Hint: There are actually two answers here.)"

I've tried using the formulas and steps I've been given to find inverses of linear equations as well as using the quadratic formula but nothing seemed to work for me and I have no clue what I'm supposed to do with this. Any help would be appreciated!

Aleguan
Nov 21, 2017

#1**+3 **

g(x)=2x^2-4x-5

For g(x), write y

y = 2x^2 - 4x - 5 factor out the 2

y = 2 ( x^2 - 2x - 5/2) divide by 2

(y / 2) = x^2 - 2x - 5/2 comlplete the square on x

(y /2) = x^2 - 2x + 1 - 5/2 - 1 factor the first three terms andsimplify

(y/2) = (x - 1)^2 - 7/2 add 7/2 to both sides

[y + 7 ] / 2 = (x - 1)^2 take both roots

±√ [ ( y + 7) / 2 ] = x - 1 add 1 to both sides

±√ [ ( y + 7) / 2 ] + 1 = x swap x and y

±√ [ ( x + 7) / 2 ] + 1 = y for y, write g^{-1}(x)

±√ [ ( x + 7) / 2 ] + 1 = g^{-1} (x)

So

g^{-1} (-2) = ±√ [ ( -2 + 7) / 2 ] + 1 = √ [5/2] + 1 or - √ [5/2] + 1

CPhill
Nov 21, 2017