Question

For the function f(x)=(3-4x)^2, find f^-1. Determine whether f^1 is a function.

Answer 1

y = (3-4x)^2
find inverse
x = (3-4y)^2

3 -4y = + or - sqrtx
4 y = 3 + or - sqrt x
y = (3/4) +/- sqrt x

this is not a function because there is more than one value of y for eaxh value of x (vertical line rule)

Answer 2

f^-1 is not a function, since f(x) sends two different elements from the domain to the same element of the range.

f must be 1-to-1 for it to have an inverse. f has two distinct branches:

y = (3-4x)^2
3-4x = ±√y
4x = 3±√y
x = (3±√y)/4

So, there is one branch
f^-1 = (3+√y)/4
and the other (3-√y)/4
where y >= 0