Chapter 1.5 - Exercises
Khan Academy - Combining Functions
ProfRobBob - Function Operations
Chapter 5.1 - Composition of Functions
Chapter 5.1 - Exercises
Khan Academy - Composition of Functions
ProfRobBob - Composition of Functions
Function Algebra, Composition of Functions
Section 1: Function Algebra
The precalculus student should be able to perform function arithmetic fairly easily, especially difference quotient which is a useful concept not only in calculus but also in business and economics, physical and biological sciences. Have students practice computing difference quotients for polynomials of degrees up to 4, rational functions and square-root functions. Students should become adept in manipulating binomial expansion of (a + b)n.
In particular, have students compute difference quotient for volumes of regular objects and compare the results with surface areas of the objects. This practice needs to be spread over the course duration to reinforce the computational aspect. Also, students become adept in finding the domain of functions.
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.
- 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
Section 2: Composition of functions
Composition of functions is a challenging concept for a precalculus student, especially in computing and finding the domain of compositions. Revisit this topic often when introducing algebraic, exponential, logarithmic and trigonometric 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.
- 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.