How would I go about this problem?

Let *F*(*x*)=*f*(*f*(*x*)) and *G*(*x*)=*F*(*x*)^2 . You also know that *f(8)=2, f(2)=3, f ' (2)=11, f '(8)=3*

Find *F*http://webwork.morris.umn.edu/webwor...100/char30.png(8)= and *G*http://webwork.morris.umn.edu/webwor...100/char30.png(8)=.

I have no idea of where to even start on this

Re: How would I go about this problem?

Use the chain rule. If $\displaystyle c(x) = a(b(x))$ for some differentiable functions $\displaystyle a,b$, then $\displaystyle c'(x) = a'(b(x))b'(x)$ by the chain rule. Do something similar to find $\displaystyle F'(x)$ and $\displaystyle G'(x)$.

Re: How would I go about this problem?

ok so how do I set up the function to find F(x)? Is it something along the lines of F(f(x), so I would use F(f(2))=h(x)?

Re: How would I go about this problem?

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Originally Posted by

**rhcprule3**

Quote:

Originally Posted by

**rhcprule3** ok so how do I set up the function to find F(x)? Is it something along the lines of F(f(x), so I would use F(f(2))=h(x)?

$\displaystyle F'(x)=f'(f(x))f'(x)~\&~G'(x)=2F(x)F'(x)$

Now you are give all necessary information to complete the substitutions.

Re: How would I go about this problem?

ok but what do I do with the given functions? like for f(8)=2, do I just plug in f(8) for f(x)?

Re: How would I go about this problem?

Like 2(f(8)8(f(3)? and i have no clue how to get G'(x). Id rather be walked through the problem so I can learn step by step how to do this problem...

Re: How would I go about this problem?

Quote:

Originally Posted by

**rhcprule3** Like 2(f(8)8(f(3)? and i have no clue how to get G'(x). Id rather be walked through the problem so I can learn step by step how to do this problem...

Well, it is evident to me that your misunderstandings are so profound that the help you need is beyond what we can give. You need to sit down with a live instructor and discuss what you can do to strengthen your understanding of the background material necessary for being successful in this course.

Re: How would I go about this problem?

Quote:

Originally Posted by

**rhcprule3** Like 2(f(8)8(f(3)? and i have no clue how to get G'(x). Id rather be walked through the problem so I can learn step by step how to do this problem...

Exactly as Plato said, you seem to have some fundamental knowledge missing. But I disagree with him about the level of help we can be. It seems like you don't understand how to use functions.

A function is like a machine. You give it input, it does something, and it gives you output. So, the notation $\displaystyle f(x)$ means if you give the function the input $\displaystyle x$, you get the output $\displaystyle f(x)$. So, the value of $\displaystyle f(x)$ when $\displaystyle x=8$ is just $\displaystyle f(8)$. If you have $\displaystyle F'(x) = f'(f(x))f'(x)$, then you are asked to find $\displaystyle F'(8)$, you just put 8 wherever you see an $\displaystyle x$ in $\displaystyle f'(f(x))f'(x)$.

Now, how do you evaluate the result $\displaystyle f'(f(8))f'(8)$? You use the information you are given. You are told f(8)=2, f(2)=3, f ' (2)=11, f '(8)=3. When you have an expression like $\displaystyle f(8)=2$, it means that if you give the machine $\displaystyle f(x)$ the input 8, you get back the output 2. So, $\displaystyle f(x)$ is a placeholder for a number. When you put a number into $\displaystyle x$, you get back a number. It is no longer a placeholder. $\displaystyle f(8)$ is a number. One of the rules of functions is that each time you give it the same input, you get the same output. So, you will never find that $\displaystyle f(8)=2$ the first time you plug in 8, then the next time you plug in 8, you get a different number. Once you know $\displaystyle f(8)=2$, you can consider $\displaystyle f(8)$ and $\displaystyle 2$ to be the same number.

So, suppose you have $\displaystyle f(a) = b$ and you are asked to evaluate $\displaystyle f(f(a))$. That would be $\displaystyle f(f(a)) = f(b)$.