Question

prove that any linear function is bijective

Answer 1

okay im srry what the bijective mean

Answer 2

bijective means it is surjective and injective

Answer 3

I'm not quite sure how to structure the proof so that it shows that it's one-to-one and onto

Answer 4

continuous *

Answer 5

oh structuring the proof is the task at hand lol

Answer 6

i guess we need an example
f(x)=ax+b

Answer 7

formal definition of injection for any d,g in R f(d)=f(g) means d=g
let's show this for f(x)=ax+b

Answer 8

start: f(d)=ad+b , f(g)=ag+b
f(d)=f(g) implies ad+b=ag+b ==> ad=ag then it is clear that d=g
f(x)=ax+b is one to one for any x in the domain

Answer 9

now comes onto

Answer 10

f is onto if there is an e in R such that f(e)=l
l has to be in the range of f but for linear function it is just R

Answer 11

so start: f(e)=ae+b <== we assign this a value l
since this maps reals to reals ( there is not range restrictions), so f must be onto

Answer 12

it is then that f(x)=ax+b is one to one correspondence
this is for any linear equation

Answer 13

do you get it?

Answer 14

Hmm, I understand the one-to-one but I'm still having trouble with the onto.

Answer 15

yeh kinda sloppy argument
the onto hinges on the fact that the elements of range (codomain) must not bet left out
they all have to be mapped to elements of domain