Express the function in the form

f compose g.
(Use non-identity functions for f and g.)
u(t) = cot t/(9+cot t)

{f(t), g(t)}=?

To express the function u(t) = cot t / (9 + cot t) in the form f compose g, we need to find non-identity functions for f(t) and g(t) such that u(t) = f(g(t)).

Let's start by finding a suitable function g(t):

We can choose g(t) = cot t.

Now, let's find the function f(t) such that u(t) = f(g(t)):

We have u(t) = cot t / (9 + cot t).

Let's substitute g(t) into the expression for u(t):

u(t) = f(cot t).

Comparing this with the given function u(t), we can see that f(t) should be equal to t / (9 + t), as f(cot t) should be equal to cot t / (9 + cot t).

Therefore, the function expressed as f compose g is:

{f(t), g(t)} = {t / (9 + t), cot t}.

To express the function in the form f compose g, we need to find two non-identity functions f(t) and g(t) such that u(t) = (f compose g)(t). Let's break down the process step by step:

Step 1: Identify the inner function, g(t).
The inner function, g(t), will be the function that is applied first.

Step 2: Identify the outer function, f(t).
The outer function, f(t), will be the function that is applied second, after the inner function g(t) is applied.

Step 3: Express u(t) in terms of g(t).
Since u(t) is already given, we can identify u(t) in terms of g(t) as u(t) = g(t) / (9 + g(t)).

Step 4: Express g(t) in terms of t.
To express g(t), we rearrange the equation from Step 3: g(t) = u(t) * (9 + g(t)).

Step 5: Identify f(t) based on g(t).
Since f(t) is the outer function, we can express f(t) in terms of g(t) as f(t) = g(t).

Now that we have all the steps, we can determine f(t) and g(t):

f(t) = g(t) = u(t) / (9 + u(t))

So, in the form f compose g, the functions are:
f(t) = g(t) = u(t) / (9 + u(t))

since cot t appears several places, let g(t) = cot(t)

Now we have

u(t) = g/(9+g)

So, let f(t) = t/(9+t) and you have what you want.

f◦g = f(g) = g/(9+g) = cot t/(9+cot t)