# Thread: Testing if an inverse exists

1. ## Testing if an inverse exists

The question:
Show that the function f: R -> R, given by f(x) = x^2, is not one-to-one.

My attempt:
According to my text, "A function f is said to be one-to-one if f(a) = f(b) implies that a = b whenever a, b is an element Dom(f).

So:

$\displaystyle a^2 = b^2$
$\displaystyle \sqrt{a^2} = \sqrt{b^2}$
$\displaystyle |a| = |b|$

I know by intuition that x^2 is not one to one, however it appears that a does equal b. Or is the fact that each side is the absolute value that the proof fails? Thank you.

2. Originally Posted by Glitch
The question:
Show that the function f: R -> R, given by f(x) = x^2, is not one-to-one.

My attempt:
According to my text, "A function f is said to be one-to-one if f(a) = f(b) implies that a = b whenever a, b is an element Dom(f).

So:

$\displaystyle a^2 = b^2$
$\displaystyle \sqrt{a^2} = \sqrt{b^2}$
$\displaystyle |a| = |b|$

I know by intuition that x^2 is not one to one, however it appears that a does equal b. Or is the fact that each side is the absolute value that the proof fails? Thank you.
The fact that, for example, f(-1) = f(1) = 1 clearly demonstrates that f is not 1-to-1.

3. I know that it isn't one-to-one, I was just trying to apply the formal procedure outlined in my text.

Now that I think about it, the absolute value does break it since it makes different signed values for a and b equal even if they're not (such as your example). Thanks.

4. Yes, the crucial point in this problem being that "|a|= |b|" is NOT the same as "a= b"!