Math Is Fun Forum

ABC1234

Member

Registered: 2016-12-22

Posts: 20

Functional Equations

Please help on the following:

1) Suppose that f(x) and g(x) are functions which satisfy f(g(x)) = x^2 and g(f(x)) = x^3 for all x >= 1. If g(16) = 16, then compute

. (You may assume that f(x) >= 1 and g(x) >= 1 for all x >= 1.)

2) The function

satisfies

for all real x. Find f(x).

3) Suppose we have the following identity:

Find the minimum of

Thanks for the help!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Functional Equations

Hi;

---

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: 10,627

Re: Functional Equations

hi ABC1234

Q2.  If you replace x with (1-x) throughout in the given equation, you can use both equations to eliminate f(1-x) and hence get an expression for f(x).

Still thinking about Q1.

LATER EDIT:

Let y = log (g(4))  then g(4) = 2^y

Evaluate f(g(4))  and g(f(2^y))

Using the given g(16) = 16 you can work out y.

Bob

Last edited by Bob (2017-04-12 06:53:01)

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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

ABC1234

Member

Registered: 2016-12-22

Posts: 20

Re: Functional Equations

for question 2 I got

. Is that right?

For Q1 I'm getting that y=4/3

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Functional Equations

Q1.  That's what I got too. 

Q2.  Oh.  So I've actually got to do the algebra.    OK give me 5 mins or so.

BACK AGAIN. Took a bit longer because I got a slightly different result.  So I tested each by setting x = 2 and evaluating the original equation left and right to see if that value of x gave the same result for both.  Sorry to say mine worked and yours didn't.

So I think there's one sign mistake in your answer.  I'll leave you to find it.

Bob

Last edited by Bob (2017-04-12 21:05:14)

---

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

ABC1234

Member

Registered: 2016-12-22

Posts: 20

Re: Functional Equations

for Q2 I got

but it wasn't right...am I still doing something wrong?

UPDATE:
Never mind, I just had to simplify it

Last edited by ABC1234 (2017-04-13 10:16:16)

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Functional Equations

Last edited by thickhead (2017-04-13 19:48:35)

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Functional Equations

hi ABC1234

That expression looks better.    And it cancels down to thickhead's answer.  So we are all in agreement.

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

ABC1234

Member

Registered: 2016-12-22

Posts: 20

Re: Functional Equations

I also need some help on this problem:
Find all functions

that satisfy

for all nonzero x.

I know I'm supposed to substitute (x-1)/x for x, but I don't seem to be getting anywhere with that. Thanks in advance!

zetafunc

Moderator

Registered: 2014-05-21

Posts: 2,436

Website

Re: Functional Equations

Has the uniqueness of the solution to (2) been justified?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Functional Equations

I also need some help on this problem:
Find all functions

that satisfy

for all nonzero x.

I know I'm supposed to substitute (x-1)/x for x, but I don't seem to be getting anywhere with that. Thanks in advance!

Being a homework problem I can give only hint.
In the previous problem you went from A to B and directly returned to A. Now try going from A to B , B to C and from there to A.

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Functional Equations

Has the uniqueness of the solution to (2) been justified?

Oh?  Am I missing something here?  The substitution x --> 1-x is valid for all real x.  The pair of simultaneous equations have a unique solution; leading to f(x) as a quartic over a quadratic.  The cancelling is valid providing the quadratic is not zero; which happens at two values.  But f(x) is continuous through both so it's even ok then.

So I'll be most surprised when you supply a second function for f(x).

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

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Functional Equations

(1)  My answer is 4/3.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}