Calculus College

Question

1.Express the function 𝐻(𝑥)=^7√x/(3+x) as f ∘ g .

f(x)=?

g(x)=?

2.Suppose 

𝑓(𝑥)=sin(𝑥),

𝑔(𝑥)=𝑥/(𝑥−4),

ℎ(𝑥)=(𝑥+7)^3.

Find 𝑓∘𝑔∘ℎ.

(𝑓∘𝑔∘ℎ)(𝑥)= ?

Answer 1

1.

There are many solutions.

The simplest is

f(x) = H(x)

g(x) = x

Perhaps a more interesting one is

f(x) = ^7√(x-3)/x

g(x) = 3+x

2.

Perhaps things will be clearer if we rename the variables

f(u) = sin(u)

g(v) = v/(v-4)

h(x) = (x+7)^3

Then g°f(x) is obtained by replacing v with f(x), i.e.,

g°f(x) = (x+7)^3/((x+7)^3)-4)

Similarly f°g°h(x) is obtained by replacing u with g°h(x).

So

f°g°h(x) = sin( (x+7)^3/((x+7)^3)-4) )