Composition of Functions

 

The composition of functions is a process by which new functions are created or “composed.” This process happens as we replace all instances of the variable X with another function. This function can be of the variable X or of another variable altogether. The following symbolic mathematical language is often used in the composition of functions:

 

(f º g )(x) or f(g(x)

 

In fact, these two expressions are equivalent, when we define f º g to mean “f composed with g.” What symbolic expression would you write for “g composed with f?”

 

Let’s look at some examples. Suppose f(x) = 3x – 8  and  g(x) = x2 + 7, and we want the newly composed function of

 

(g º f)(x). One way to remember the order of composition is this: the function on the “inside” of the statement replaces X in the function on the “outside” of the statement. So, using this analogy, f goes into g. Here’s how:

 

 

g(x) = x2 + 7        and    f(x) = 3x - 8

 

 

          (g º f)(x) = g(f(x)) = (3x – 8 )2 + 7 = (9x2 – 48x + 64) + 7   = 9x2 – 48x + 71, which is a quadratic function.

 

Now, suppose we need to know (f º g )(– 6).  First find f º g. Then substitute – 6 for X in the new function:

 

 

3(x2 + 7) – 8  = 3 x2 + 1  Now, evaluate the function at – 6:   3(– 6)2 + 1 = 109

 

 

And, there you have itècomposition of functions! As we saw in class, there are worthy applications of composition of functions.

 

 

 

Application of Composed Functions

 

 

A lakeside city, whose economy depends largely on the lake, is concerned about the presence of a certain pollutant (concentration in parts per million or ppm). Scientists have developed a model directly proportional to the population of the city:  C(p) = 2.54 p + 98.55.

 

The city government has also developed a model for the growth (in thousands of people) of the city: P(t) = 1.47t + 27.8.

 

Scientists say that the pollutant concentration of 410 ppm is harmful to vegetation and wildlife. For how many years can the city expect no major problems, based on continuing trends and the models they have available?

 

 

Solution

 

          The city government needs to know how long it will take the pollutant to rise in concentration levels to reach a harmful level for vegetation and wildlife around the lake. In terms of variables, then, they need the concentration level over time. Notice that the models available show concentration in terms of population and population in terms of time. This situation is the classic case for composed functions. It also shows the situation where two functions are in different variables, as I mentioned earlier.

 

          What mathematics is involved here? To solve the city government problem, we must compose P into C, and we can do this because we have an equivalent function for P.

 

 

C(p) = 2.54 p + 98.55  and  P(t) = 1.47t + 27.8

 

         

 

Now replace P with the equivalent function:                      C(t) = 2.54 (1.47t + 27.8) + 98.55                   Notice the change in the newly composed function.

 

 

Simplifying, we get:                                    C(t) = 3.73t + 169.16,  which will tell us how long the city has to work on the pollutant problem before serious damage may be

                                                                                                      Done to the local economy.

 

 

There are two ways to find the amount of time: 1) set the equation equal to 410 ppm and solve for t, or 2) put the equation in Y1 of the calculator, graph it, and use the TABLE function to find the value.  The number of years before the city must find a solution is 64 years.