Step One For convenience f(x) = y y = -(1/2) sqrt(x + 3) Interchange x and y x = -(1/2) sqrt(y + 3) Multiply both sides by -2 (-2x) = sqrt(y + 3) Square both sides 4x^2 = y + 3 Subtract 3 from both sides 4x^2 -3 = y f-1(x) = 4x^2 - 3
Graphs The original graph is in blue The inverse is red. Green Line y = x : explained below.
Comment Inverses have a very strange property. It acts as a check.
If you perform the following operation f-1(f(x)) you are left with x
In this case f-1[-1/2 sqrt(x + 3)]= 4(-1/2 * sqrt(x + 3))^2 - 3 f-1 (-1/2sqrt(x+3)) = 4 (1/4 (x+3) - 3 f-1(-1/2sqrt(x + 3)) = x+ 3 - 3 = x
I've included this because I think you have to put the same restriction on f-1(x) as you do on f(x) which is that for f(x) x ≥ -3
This last assertion could be incorrect, but I don't think so. The answer is f-1(x) = 4x^2 - 3 <<<<< answer first box and x≥-3 for the second box. <<<<<< answer for the second box.
Let me know if these are correct. I'm not sure about the restrictions.