Re: (Very Off Topic): Need Help With Linear Maps Question

[Dave L. Renfro]
Problem on test: Prove that f(x) = 2x - 1 is a 1-to-1 function.

Actual student answer: The function takes 1 to 1, and therefore
it's a 1-to-1 function.

[David T. Ashley]
Just out of curiousity (I'm not a mathematician), what is a valid
proof that this is 1:1?

Start with the definition. Stop there too :-)

"f is 1-to-1" means:

f(x) = f(y) implies x = y

or, equivalently,

x =/= y implies f(x) =/= f(y)

(read "=/=" as "does not equal"). The first version is usually easiest to
use in a proof. In this case,

f(x) = f(y) implies [plugging in f(z) = 2z-1]
2x-1 = 2y-1 implies [adding 1 to both sides]
2x = 2y implies [dividing both sides by 2]
x = y

So f satisfies the definition of 1-to-1.

I'm tempted to say that if g(x) is the inverse defined as (x+1)/2,
then for any real number x you get:

g(f(x)) = ((2x-1) + 1)/2 = x, i.e. you always get the same result
back.

This seems to prove 1:1, since if it were not 1:1 and both f() and
g() are functions there would be some "non-reversible" value for
the argument.

Showing that the inverse function is well-defined works too, since f is
one-to-one if and only if f has an inverse defined on f's codomain. As
above, it's usually easier to work directly with the definition.

But how would a mathematician do this proof? (I'm just a hack.)

Ah, a /mathematician/ would say "it's obvious" and move on to the next post
;-)


.

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