1. Section 2.4 Composite and Inverse Functions

2. Consider the following situation
There is an oil spill and the oil is disseminating from the source in a circular fashion
We take some measurements and notice that after one second a particular point at the edge of the slick is 5 feet from the source, at 2 seconds that same point is 10 feet from the source, at 3 seconds, it is 15 feet from the source
What we need is to develop a function that will give us the area of the oil slick in terms of time
What formulas may be useful here?
Come up with a formula for the area of the slick and the point seperately

3. We could write A = f(r(t)) = g(t)
When the output of one function is the input for another function, the two functions form a composition
In the previous example the function giving the radius as a function of time became the input for the function that gave the area as a function of the radius
We could write A = f(r(t)) = g(t)
We now have that area is a function of time
We essentially cut out the middle man

4. Come up with a formula for P = f(t)
The following table gives the population, P, of the town of Jonesville as a function of time, t, where t is the number of years since 1980
Come up with a formula for P = f(t)
Tell in what year the population reaches 30,000
Can we always find the year given the population? t 1 2 5 10
P = f(t)
15,000
15,200
15,400
16,000
17,000

5. Do we still have a function?
Now if we swap the independent and dependent variable, we get the following table
Do we still have a function?
Find the inverse function, f -1(P) for this problem
Let’s see what happens when we compose f and f -1
Turns out if f and are inverses we have P
15,000
15,200
15,400
16,000
17,000
t = f -1(P) 1 2 5 10

6. In your groups work on problems 1, 7, and 17
The revenue, R, in thousands of dollars for selling x units of a given product is given by R = f(x) = 500x
Find and interpret
f(10)
f -1(1000)
f -1(R)
In your groups work on problems 1, 7, and 17