Function Algebra, Composition of Functions

Function Algebra

Function arithmetic and algebra deals with manipulating functions using the usual arithmetic of numbers. You may recall that arithmetic operations on numbers include sum, difference, multiplication, division and exponentiation. A particularly important computation required in calculus is that of the difference quotient of a function (also known by slope of a line, average rate of change, etc.) over an interval. In addition to these operations, we also learn about composition of two or more functions in a later section.

The result of arithmetic operations on functions is again a function. So naturally we would be particularly interested in the domain of the resulting function which in turn depends on the individual functions. 

Learning Objectives:

  • Function arithmetic - Sum, Difference, Product, and Quotient
  • Composition of Functions
  • Domain of Resulting Function
  • Difference Quotient for polynomials, rational functions and nth-root functions. 
  • Examples of functions in practice -
    • Position function of a projectile
    • Compound Interest
    • Volume of Regular objects (cylinder, cone, sphere)

Price-Demand, Revenue Cost, Average Cost and Profit function

Composition of functions

In this module we will learn about compositions of two or more functions where the output value of one function becomes the input value for another function. The concepts are used in the study of inverse functions and other algebraic functions.

Learning Objectives:

  • Find the composition of two functions.
  • Evaluate a composition.
  • Perform function arithmetic with compostion including average rate of change and difference quotient
  • Determine the domain of composition of two functions.
  • Identify the functions whose composition in a specific order is a given function.