Question

Here's a somewhat "problem-solving" type question I was doing earlier on today in a written math contest. I don't recall the question in verbatim but remember it pretty well. Whoever solves first gets a medal :D

"Find all linear functions \(\bf f(x)=ax+b\), if \(\bf g(x) = f^{-1}(x)\), then \(\bf f(x) - g(x) = 44\)."

Answer 1

well f(x) = ax+b is only linear if b = 0:)
that should say find all affine functions

Answer 2

@zzr0ck3r Remember I don't recall the question in verbatim. Also, the wording of the question implies that f(x) is a linear function hence a and b are constants...Also the question implies the invertibility of f(x) hence a cannot be 0. I shouldn't be telling you this but whatever lol :D

Answer 3

was more a joke. Im sure it says linear...very common thing.

Answer 4

b cant be constant

Answer 5

unless its 0
linear must go through the origin

Answer 6

@zzr0ck3r when I say "constant", I don't mean that b is fixed...

Answer 7

this is a trick. he is taking a test and doesnt want it to bee obvious. also, he is an retriceole.

Answer 8

No he aint

Answer 9

retricehole

Answer 10

A S S H O L E

Answer 11

@tylerj37 I already solved this problem this morning before going to bed. just wanted to see who solves it first here on openstudy. I did it in about 5-6 minutes I think...with the writing and stuff combined

Answer 12

Im saying @genius12 that y=mx+b is only linear if b=0. A linear equation must go through the origin.

Answer 13

y = ax + b so the inverse is y = (x-b)/a

\[\Large ax + b -\frac{ x-b }{ a }= 44\]

Answer 14

most people call affine equations linear when they are not.

Answer 15

@zzr0ck3r The question says "Linear function". In the current context, a linear function is defined to be a polynomial of degree 0 or 1. Hence b is not required to be 0.

Answer 16

@agent0smith That would be the first step. Keep going.

Answer 17

well that's wrong:) a linear function is what it is...there are rules:)

Answer 18

@zzr0ck3r You're wrong. Sorry to say.

Answer 19

go read man....

Answer 20

do you know what a linear transformation is?

Answer 21

how then can mx+b be a linear transformation?

Answer 22

http://math.stackexchange.com/questions/275310/what-is-the-difference-between-linear-and-affine-function
just google it:)

Answer 23

its not important...again I was just making a joke as it is a very common misconception

Answer 24

@zzr0ck3r You are referring to linear algebra. A linear function - a linear map - is different from a linear function in calculus and related areas.

Answer 25

no its not

Answer 26

yes it is go read

Answer 27

I don't need to read, my analysis teacher had this discussion in front of my class 3 days ago

Answer 28

show me anything that says what you are saying....

Answer 29

That's great lol. Doesn't prove you right.

Answer 30

again, one thing.

Answer 31

just one

Answer 32

:)

Answer 33

http://en.wikipedia.org/wiki/Linear_function

Answer 34

now please don't say that wikipedia isn't a scholarly source lol...

Answer 35

lol

Answer 36

ok man

Answer 37

its the definition of linear

Answer 38

why would it be different for calculus...

Answer 39

lol

Answer 40

there is lots of terms used differently in various fields of mathematics. not each term is defined the same way in every field.

Answer 41

again very common misconception...

Answer 42

http://mathmistakes.info/facts/AlgebraFacts/learn/ctm/nlin.html

Answer 43

ok you need to stop with the misconceptions. lol. this is ain't no misconception. =]

Also that link doesn't help prove your argument..

Answer 44

you cant prove notation, it is what it is by definition

Answer 45

you give me the definition that wiki gives you, ill stick to my analysis book.

Answer 46

like i said, what subject/field is your "analysis" book for

Answer 47

pick one

Answer 48

dude I just showed you a site that says common mistakes....and then goes on to list the exact thing im talking about first. I know its not proof, but you might want to go read more than wiki about it.

Answer 49

again, was not trying to start anything. just something I say when people say linear.

Answer 50

sort of like when people say derive when they mean differentiate. I cant help it.

Answer 51

@zzr0ck3r if you were making a joke, how about not sh*tting up this question with it? Start a new question on the definition of a linear function and go nuts there.

\[\Large x(a-\frac{ 1 }{ a}) + b + \frac{ b }{ a } = 44\]
a-1/a = 0, and b+b/a =44

Answer 52

ps. calculus is mostly performed on vector spaces, there is no seperation

Answer 53

like the real numbers...

Answer 54

a =1

b = 22

Answer 55

I can post more working, but the thread's already a mess.

Answer 56

we are smart, we can read around comments.

Answer 57

@zzr0ck3r The author of that page calls linear functions in calculus as affine functions and says that the term "linear functions" should be defined the way it is liner algebra to avoid confusion. Once again, he's just saying that to avoid confusion. That in no way however proves to say that a linear function cannot be defined the way it is in calculus. Now in the future, a math conference may conclude that the term "linear function" will be replaced with the term "affine function" but for now, we are sticking to how a linear function is defined in the current context.

@agent0smith Your answer is correct. But how many such f(x) that satisfy the conditions are there then?

Answer 58

Only one that i found, f(x) = x + 22

a=-1 isn't a valid solution.

Answer 59

again you are acting like linear algebra and calculus are independent of each other.

Answer 60

@agent0smith Good job. There is a single solution y = x + 22. That's end of this thread. Cheers.

Answer 61

lol