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(8)= and G(8)=.
I have no idea of where to even start on this
A function is like a machine. You give it input, it does something, and it gives you output. So, the notation means if you give the function the input , you get the output . So, the value of when is just . If you have , then you are asked to find , you just put 8 wherever you see an in .
Now, how do you evaluate the result ? 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 , it means that if you give the machine the input 8, you get back the output 2. So, is a placeholder for a number. When you put a number into , you get back a number. It is no longer a placeholder. 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 the first time you plug in 8, then the next time you plug in 8, you get a different number. Once you know , you can consider and to be the same number.
So, suppose you have and you are asked to evaluate . That would be .