1- If we have:

**f(x)= x ^{^2}+2x**

**g(x)=x ^{^3}**

^{ }

Then, how many answers can **(gof)(x)=(fog)(x)** have?

1- **1**

2- **2**

3- **3**

4- **6**

******* ******* ******* ******* *******

2- If we have:

**f(x)=2[x]+2[-x]**

** **

Then, what is the set of **fofof(x)**?

1- **{2}**

2- **{0}**

3- **{0,4}**

4- **{0,-2}**

******* ******* ******* ******* *******

3- If we have:

**f(x)= (1-x)/(1+x)**

** **

Then what is **fof…f(x)**?

*P.S.**In fof…f(x), f is repeated 100 times*

** **

1- **X**

2- **1/x**

3- **(1-x)/(1+x)**

4- **{(1-x)/(1+x)}^ ^{100}**

## Replies

A physics lover!

Very nice.

I lik

edit too!Good job!

All of them are right except the first.

It seems that you are so good at math!

Please tell us about your ways that you are using.

We like to know about your logic too.

Please let me write their solutions here:

1.

fog(x)= f(g(x))= f(x^3) =(x^3)^2 + 2 (x^3)= x^6 + 2(x^3) (i)

gof(x) = g(f(x)) = g(x^2 + 2x) = (x^2 + 2x)^3 (ii)

(i) = (ii) => x^6 + 2(x^3) = (x^2 + 2x)^3

=> x^3{(x+1)^2} =0

x=0 or x =-1

It has

2 answers.******* ******* ******* ******* *******

2.

f(x)=2[x]+2[-x]fofof(x) = f(f(f(x)))

f(x) is always exact number (I mean without any decimal amount).

And also for x which belongs to "Z", f(x) will be

0.Then f(f(x))=0

=>f(f(f(x))) = f(0)=

0******* ******* ******* ******* *******

3.

fof(x) = f(f(x)) = f((1-x)/(1+x)) = x

fofof(x)= f(f(f(x)))= f(x)

fofofof(x) = f(f(f(f(x))))= f(f(x)) = x

...For

odd repeatedit isf(x), and foreven repeatedit isx.100 is even, then it is equal to

x, namely:fof…f(x)= 0(When f is repeated 100 times)2 or 3 minutes!

Very good!

I'm so impressed!!

I think you love it :)

Can I ask how much time it took that you solved it?!

I think it took less than few seconds!

You are very good at math!

I like your easy ways for these not easy questions so much!!

Thank you for sharing your solutions.

WoW!

Sue!!

Well done, dear!

Now, all of them are right!

Very good!

But, which ways are you using?!