Determine whether the function has an inverse function. : y=lnx^2

Question

anonymous · Answer

$$\ln(x^2)~~\text{or}~~(\ln x)^2~~?$$

anonymous · Answer

It doesnt have any perenthesis

anonymous · Answer

If that's the case, I'd assume the first one.

anonymous · Answer

|dw:1377745104389:dw|

anonymous · Answer

The usual procedure is to flip the variables, then solve for $y$.
$$y=\ln(x^2)~~\Rightarrow~~x=\ln(y^2)$$

anonymous · Answer

got that

anonymous · Answer

but how fo you solve for y?

anonymous · Answer

$$x=\ln(y^2)\
x=2\ln y\
\frac{x}{2}=\ln y\
y=\cdots $$

anonymous · Answer

In my book, the answer says there is no inverse... Im assuming that's because if you graph the function it doesn't pass the horizontal/one-to-one test
But how would I know that by solving it algebraically like that 
I have no idea how to graph an equation like that either

zepdrix · Answer

https://www.desmos.com/calculator/fw7vmht6fc Oh I see what you mean. Look at the graph I provided. Our red function, when reflected over y=x (in green) gives us the inverse function in purple. See how we lost one of the branches? :O (Otherwise it wouldn't pass the horizontal line test, as you said.) Algebraically? Hmmm..