If f(x)=x+4 and h(x)=4x-1, find a function g such that g(f(x)) = h(x).

We get the above answer by noting that g(x) must be linear
otherwise when applied to (x+4) we would not get another linear
function.

So suppose
g(x)=ax+b

g(f(x))=g(x+4)=a(x+4)+b=ax+4a+b

Now
we are told that
g(f(x))=4x-1

So

a=4 and
4a+b=-1

4(4)+b=-1

16+b=-1

b=-17

So
our functiong g(x) must be

g(x) = 4x -
17