**Section Objectives**

- Compute sums, differences, products, and quotients of functions.
- Compute a composition of functions.
- Write a function as a composition of functions.
- Solve problems involving operations on functions.

Besides transforming individual functions, two or more functions can be combined in lots of ways to produce new functions.

The most basic combinations of functions come from the four basic arithmetic operations.

- , provided that

Given the following table of values, compute , , and .

x f(x) g(x) 0 5 2 1 -3 7 2 10 -2 3 -8 -10

- Let and . Compute . What about ?

- Sketch the graphs of and . Use the graphs to evaluate . What about ?

Another useful and important way of combining functions is to form the composition. Functions are composed when the output of one function is used as the input for a second function.

The **composition** of and , written , is the new function .

- Let and let . What function is ? What about ? What about ?

- Refer to the table above. Compute .

- Let . Find two functions and so that .

- Let and . Find and determine its domain.

- Let and . Find and completely simplify . What is the domain of the composition?

- A large spherical balloon is being inflated so that its radius, in feet, after minutes is given by . Find its volume as function of .

- An object is cooling in such a way that its temperature in degrees Celsius after minutes is given by . The formula is used to convert from temperatures in Celsius to Fahrenheit. Find the temperature of the object in degrees Fahrenheit at time .