Question

a function is linear if:

f(a+b) = f(a) + f(b)

Let f(x) = 2x+3 ; is this linear?

Answer 1

im curious if we would have to move this to pass thru the origin for this definition of work out

Answer 2

does the constant at the end matter?

Answer 3

f(x) = 2x+3

f(1) = 2(1)+3 = 5
f(2) = 2(2)+3 = 7
f(3) = 2(3) + 3 = 9

f(3) = f(1+2) = f(1) + f(2) = 5 + 7 = 12

nah, this is not same like the defenition above :)

Answer 4

But @RadEn u cant generalize

Answer 5

im thinking that since it doesnt contain the point (0,0); since its of the family 2x+ k,

Answer 6

f(x)+f(x+3)=4x+12, f(2x+3)=4x+9

Answer 7

2(a+b) = 2(a) + 2(b)

Answer 8

im not sure of its proper to say that {2x+k} is a congruence class or not

Answer 9

congruence class??

Answer 10

Congruence class? Does it have anything to do with the linearity of a function?

Answer 11

a congruence class is a set of elements that are by all rights ... equaivalent

like 7 mod 6 = 13 mod 6 because 7 and 13 are both equal to 1 mod 6

they are in the same congruence class mod 6

Answer 12

if all functions of the form: {2x + k} are in the same congruence class, then it would most likely be proper to say that f(x) = 2x+3 is linear, since f(x) = 2x+0 is linear

Answer 13

Why not just use the defining property of a linear function, which was already given in the problem?

Answer 14

becasue:

2(a+b) + 3 is not equal to 2a + 3 + 2b + 3

Answer 15

but ... since the line 2x+3 and 2x are parallel, wouldnt the properties of 2x be the same as 2x+3?

Answer 16

barring of course the y intercept

Answer 17

Yes, I have the same concern, but if the definition says it's not linear, then it's not.

Answer 18

I think the vector space section explains it pretty well: http://en.wikipedia.org/wiki/Linear_function

Answer 19

but im sure that definition is based on .. i forget the name, kernel maybe? ... such that 2x+3 is not a proper vector space (or someting) since it doesnt contain the (0,0) point

Answer 20

pfft, i can never read those wiki with any degree of sensibility :)

Answer 21

Then go search linear function vs. affine function. Basically my intuition was just wrong, a function f(x)=ax+b is called a linear function iff b=0, though its graph is still line-ar.

Answer 22

i thought about googling it later today, just thought id bring it up in here for people to be able to ponder something other than ... well, most of whats posted here

Answer 23

affine is something i need to get better acquainted with fer sure

Answer 24

Consider posting it up on Math SE if you want a better insight into it.