MHB How to find composition of f ° g and g ° f ?

[Henry R]

Let f and g be function from the positive integer defined by the following pairs f : N →R,g∶N →R. Find the composition f ° g and g ° f.A)

F(n) = log(n+3)

g(n) = log log(n + 10)
....................
B)

F(n) = n log(n+2)

g(n) = n^(1.1)
....................

Could you guys give me answer? Anybody? Please..

Here is my answer for the other question...

f(n)= n + 60 g(n) = 100n

and my answer is this ;
f ° g = 100n + 60

g ° f = 100 ( n + 60 )
= 100n + 6000.

So, can you guys answer the question of A and B?

 

[Evgeny.Makarov]

Let f and g be function from the positive integer defined by the following pairs f : N →R,g∶N →R. Find the composition f ° g and g ° f.

If a factory $X$ converts iron ore into iron and you want to use that iron somewhere else, you would need to find another factory that takes iron as input. It won't work to find a factory $Y$ that converts cotton into clothes and try to combine them as in $Y\circ X$. Similarly, you can't feed the output of $f$ into $g$ because $f$ outputs real numbers, such as $\pi$ and $e$, and $g$ expects natural numbers as input.

I assume that in $f\circ g$ the function $g$ is applied first. This is the most prevalent convention, and your example with $f(n)= n + 60$ and $g(n) = 100n$ follows it. However, there are some sources that use the opposite order.

If $f:\Bbb R\to\Bbb R$ and $g:\Bbb R\to\Bbb R$, then the rule of finding $f\circ g$ is the following. Write the definition of $f(x)$ and replace $x$ with the definition of $g(x)$. By definitions I mean the right-hand sides of equalities defining $f$ and $g$. So, in A) the definition of $f(x)$ is $\log(x+3)$ and the definition of $g(x)$ is $\log\log(x + 10)$. Replacing every occurrence of $x$ in the first expression by the second expression, we get $\log(\log\log(x + 10)+3)$. Now I suggest you try finding $g\circ f$.

Note also that in mathematics lowercase and uppercase letters may denote different entities, so $f$ and $F$ may well be different functions and should not be confused.