what is the inverse of f(x) = 3x^2 + 4?

Question

anonymous · Answer

it has no inverse ... unless you restrict the domain

anonymous · Answer

to give it a shot ... let x = $f^{-1}(x)$

anonymous · Answer

thank you!

anonymous · Answer

its B

anonymous · Answer

$$ f(x) = 3x^2 + 4$$
$$\large  \underbrace{f(f^{-1}(x))}_{\text{=x}} = 3(f^{-1}(x))^2 + 4$$
$$x = 3(f^{-1}(x))^2 + 4$$
now algebra out the inverse

mathmale · Answer

"algebra out the inverse" could be better expressed as "solve the above equation for $$f ^{-1}(x).$$"

anonymous · Answer

lol :)  perhaps

mathmale · Answer

There are complications here in that finding a square root produces two "answers."  How do we know which one is the correct one?  or would both represent the inverse of the original function?

In either case, the inverse function has a restricted domain.  What is that restriction?

mathmale · Answer

@guest.001 's solution, in which he begins by writing $$x=f(f ^{-1}(x)),$$ is both interesting and valid, although most algebra textbooks present a slightly different method:

1) replace f(x) by y
2) interchange x and y in the resulting equation
3) solve the resulting equation for y
4) replace that y by $$f ^{-1}(x)$$

Note:  we'll still have two possible "answer" choices from which to choose, and the domain of the inverse function is restricted in the same way as it was using guest.001 's solution.