Do these functions have an inverse function?

Question

A. y=ln x^2 (ln=natural log)

B. y= 2^(3-x)

C. y= x^2 - 4x+6

How do you figure it out?

For letter A. the answer is "no" but I don't know how to get that.

Answer 1

rewrite as x = ... y's
then swap x and y.

A) y = ln(x^2)
y = 2ln(x)
y/2 =ln(x)
thus x = e^(y/2)
so the inverse is f(x) = e^(x/2);
(some details for x not 0)

Answer 2

The function has an inverse if it's one-to-one, ie. for every value of x you have one value of y, and there are no two x's which have the same y.

If the inverse exists, you can find it by rearranging and expressing x as a function of y. Then exchange y with x to get it to the correct format. Like this:

A: no inverse exists, as it's not 1-to-1. Proof: find two x's which have the same y. Eg. x=1 and x=-1 have the same y because of the x^2 (1^2=(-1)^2).

B: function is 1-to-1 so inverse exists. Rearrange:
y = 2^(3-x)
log{base 2} y = 3-x
x = 3-log{base 2} y = 3 - log(y)/log(2) (can be any base)
Now exchange y with x to get the inverse function:
y= 3- log(x)/log(2).

C: A quadratic expression will never be 1-to-1 as it is a parabole (=>symmetric about a vertical axis => there will be x's which have the same y!). In this case it is symmetric about x=2, so for example x=3 and x=1 give the same y value:
1^2 - 4*1 + 6 = 1 - 4 + 6 = 3
3^2 - 2*3 + 6 = 9 - 12 + 6 = 3
So it does not have an inverse.

All of these functions could have an inverse if you restricted their domain to only a certain set of x's - for example, if you restricted the domain of C to only the x's which are larger than 2, then it would be one-to-one so an inverse would exist.