# doing operations on both sides

Discussion in 'Undergraduate Math' started by G Patel, Aug 20, 2007.

1. ### G PatelGuest

In the past I've done things like taking the derivative/limit of both
sides of an equation without thinking much about why I could do this
safely.

Recently I got to thinking about "doing an operation on both sides."
Which, in general, preserve the equality after the operation? I know
there is an issue with operations like "squaring both sides" where the
new equation might have extra roots.

What type of operations can cause problems? Is there a general rule
on this?

G Patel, Aug 20, 2007

2. ### riderofgiraffesGuest

Recently I got to thinking about
In general, you want reversability.

Squaring both sides can lead to problems because
although a=b => a^2=b^2, the reverse implication
is invalid, because squaring is not reversable.
It's not one-to-one.

riderofgiraffes, Aug 20, 2007

3. ### -kk-Guest

a=b => f(a)=f(b)(obviously).
We can only assume the reverse implication [ f(a)=f(b) => a=b ]
when f is injective.

For the example you mention, f(x)=x^2: this function is not injective
when x is any real number. However, we can make any function injective
by restricting the domain (and any function surjective by restricting
the codomain). For the above example we can (for example) restrict
the domain to the nonnegative reals (or integers etc). The 'rule'
here is injectivity.

HTH

-kk-, Aug 22, 2007