Question

F ( x ) = x − 3 x , g ( x ) = x + 3 , h ( x ) = 2 x + 1 , ( g ⋅ h ⋅ f ) ( x )

Answer 1

Composing functions just means to feed the ouput of the inner one as the input for the outer one.

So, the following writings are equivalent:

$(g\circ h \circ f)(x) = g(h(f(x)))$

Using the second form, we can see that the first thing to do is to compute $f(x)$, which is given:

$f(x) = x-3$

This means that we can update the fomula:

$(g\circ h \circ f)(x) = g(h(f(x))) = g(h(x-3))$

So, now we have to compute $h(x-3)$. Given $h(x)=2x+1$, we know that $h$ doubles the input and adds one. In this case, the input is $x-3$, so we have to double this and add one:

$h(x-3) = 2(x-3)+1 = 2x-6+1 = 2x-5$

So, we have

$(g\circ h \circ f)(x) = g(h(f(x))) = g(2x-5)$

Similarly, from $g(x) = x+3$, we understand that $g$ just adds 3 to the input. In this case, the input is $2x-5$, so we'll add 3 to this quantity:

$g(2x-5) = (2x-5)+3 = 2x-2$

So, we finally have

$(g\circ h \circ f)(x) = g(h(f(x))) = 2x-2$