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How do you find $(f o g) (n)$ $(fog)(n)$ given $f (n) = 2 n$ $f(n)=2n$ and $g (n) = - n - 4$ $g(n)=-n-4$ ?

1 Answer

Oct 4, 2016

$(f o g) (n) = - 2 n - 8$ $(fog)(n)=-2n-8$

Explanation:

It is sometimes easier to thing of $(f o g) (n)$ $(fog)(n)$ as $f (g (n))$ $f(g(n))$

It might also help to replace $n$ $n$ in the definition of $f (n)$ $f(n)$ with some other variable; since $n$ $n$ is just a variable placeholder there is no problem doing this.

So $f (n) = 2 n$ $f(n)=2n$ could equally validly written as
$XXX f (k) = 2 k$ $color(white)("XXX")f(color(red)k)=2color(red)(k)$

Now, if we want to evaluate $f (g (n))$ $f(color(blue)(g(n)))$
we simply replace $k$ $color(red)(k)$ with $g (n)$ $color(blue)(g(n))$

$XXX f (g (n)) = 2 g (n)$ $color(white)("XXX")f(color(blue)(g(n)))=2color(blue)(g(n))$

and since $g (n) = - n - 4$ $color(blue)(g(n))color(black)=-n-4$
$XXX f (g (n)) = 2 (- n - 4) = - 2 n - 8$ $color(white)("XXX")f(color(blue)(g(n)))=2(color(blue)(-n-4))=-2n-8$

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