Expert: Scotto - 8/3/2010

Question
QUESTION: HI,
I'm having a really hard time with this because of the square rt. Could you  please help?
f(x)= 3sqrt 2x - 3, what is the inverse (f -1(x))
so far I have y= 3sqrt 2x - 3
         x = 3sqrt 2y -3
I know what I need to do, but I'm not sure how to deal with 3sqrt2y.  Would this be 6y?
Thanks, Cathy

ANSWER: Is the line starting with f(x)
f(x) = 3√(2x-3), what is the inverse, f^-1(x)?

Let f(x) be replaced by y, change the x and y around, and solve for y.

This means we have y = 3√(2x-3), and changing gives us to find the inverse solve
x = 3√(2y-3) for y.

First, divide both sides by 3.
Once this has been done, square all terms on both sides.
Note that 3 squared is 9, x squared is x², and the squareroot squared leaves 2y-3.

Once this has been done, add 3 to both sides.
This leaves a 2y on the right, so divide both sides by 2.
That will give the y = ...., which is the inverse of the function given.


---------- FOLLOW-UP ----------

QUESTION: The actual problem calls for the sqrt of 2x.  The -3 is outside the sqrt sign.

Answer
So when switching the x and y the problem becomes x = 3√(2y) - 3?

In this case, add 3 to both sides and get x + 3 = 3√(2y).

The next step is to divide both sides by 3, giving (x+3)/3 = √(2y).

Having isolated y by itself, take the square of each side, giving
(x+3)²/9 = 2y.

If we then divide both sides by 2, we get y = (x+3)²/18.