i have to get the inverse of f(x)=1/3 x^2 + 2x + 2. I got f-1(x)= -3 sqrt {12+12x}. But the answer key says f-1(x)= -3 sqrt {3x+3}. How did they get that?

Answer 1

So Khushi S.,

Inverse functions can be tricky. I suggest you start by actually throwing your solution for f^-1(x) into the given expression for f(x), that is, f(f^-1(x)) and expanding it to see if you really come out to just (x) -- that's what you want.

Then try that for the book solution. You can also do f^-1(f(x)), if you prefer.

You and the book solution are only different by a constant factor (both the 3's and the 12's can also be extracted from under the square root).

Just eyeballing, I don't see how that f(x) expression can be messaged so as to be f((polynomial in x)^2), so that the inverse operation involves just a sqrt.

Also, you might want to take note that the domain possible for x changes between the two functions -- so that not all values of x work both directions, so to speak. But you could figure that out from the graph of f(x): it must be a parabola, right? Therefore, the inverse function reverses the two variable axes. So look for the vertex of the f(x) parabola. That's the "starting point" for the (upper limb only! of the) f^-1(x) function, which MUST be a sqrt (since it's the inverse for a square function). The rest is just plugging in a couple of points to make sure of your offset and coefficient under the sqrt sign? And again, it doesn't look as if it is "centered" on either axis !

-- Cheers, --Mr. d.