Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

please help.

If f(x) = ax + b for all real numbers x, where a and b are numbers, find for what values
of a and b is it true that ff = f o f?

Not sure what it means by ff or f o f. Help is much appreciated.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,134

Re: please help.

ff probably means f( f(x) ) = f(ax+b) = a(ax+b) + b

Hard to say what f o f means.  Usually a 'o' in such a question would denote a binary operation which would be defined somewhere in the question.  Best guess: maybe it means f(x) squared = (ax+b)(ax+b).  Although it could also mean combine the functions as above.  In which case they'd be the same ???

The person who set this should have used proper notation then you wouldn't need to ask about it.  The whole point about functions is that they should be well defined!

Where did this come from ?

Try, for what values of a and b is              a(ax+b) + b = (ax+b)(ax+b)  for all x.

No, that's stupid because a^2x^2 would then be zero => a = 0 so f(x) = b .... => b = b^2    => b=1. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help.

Hi Bob;

That is a composition of functions.

http://www.mathsisfun.com/sets/function … ition.html

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,134

Re: please help.

Hi bobbym,

OK, so what does the other mean?

I have edited my post and annihilated most of the possibilities. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help.

Hi Bob;

ff? I have no idea other then f(x)f(x). What do you like?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,134

Re: please help.

That's my best guess.

So we have (written properly)

f( f(x) ) = f(x).f(x)  => f(ax+b) = (ax+b)(ax+b) => a.ax+ ab + b = (ax)^2 + 2abx + b^2

comparing coefficients for x^2

a^2 = 0 => a = 0

constant term

b = b^2  => b = 1

x coefficient

a.a = 2ab  OK if a = 0

So a=0;  b=1

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help.

Hi;

That looks good. Could b = 0 too?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,134

Re: please help.

b=0  Yes.  How did I miss that ?  So, a perfect answer:   the function has disappeared entirely. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: please help.

My kind of function.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.