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CSEC Mathematics Review - Introduction To Functions & Relations
The document serves as a foundational introduction to functions and relations in pre-calculus. It explains the definitions of functions and relations, types of relations such as one-to-one and many-to-one, and methods to test for functions using examples. It also discusses notation for functions, including graphical representation and the vertical line test for verifying functions.
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Introduction to functions, relations, domain, and range with basic definition and examples.
Defining functions with a machine analogy, including a currency conversion example illustrating linear functions.
Introduction to function notation and examples of functions such as f(x) = x + 1, area, and volume functions.
Explaining relations as sets of ordered pairs, including formal definitions of functions and types of relations.
Explaining one-to-one functions with examples and the concept of injective relations in mappings.
Describing many-to-one functions, including examples and criteria for these types of relations.
Clarifying that one-to-many relations do not represent functions and discussing inverse functions.
Introduction to the vertical line test to visually determine if a relation is a function with practical examples.
CSEC Mathematics Review - Introduction To Functions & Relations
- 1. Pre-‐Calculus Kevin Small – www.cxctutor.org An Introduc<on To Func<ons and Rela<ons
- 2. Topics To Be Covered • Define What is a Func/on – Show Simple Examples of Func<ons – Explain How Func<on Nota<on Works • Define What is a Rela/on – Define The Terms Domain and Range – Learn How To Draw Mapping Diagrams – Learn The Different Types of Rela<ons – Lean How To Test For Func<ons
- 3. What is a Func<on • A func<on in a typical sense is just a machine with a specific rule that produces a single output.
- 4. Example • Consider the following machine which is used to convert our local Barbadian currency into US Dollars. OR y = 0.5 x (5)BBD Currency Converter (2.5)USD Output Rule Input
- 5. Example Con<nued The func<on given is an example of a Linear Func/on. We will discuss the graphs of func<ons later but here is the graph of our currency converter func<on.
- 6. Func<on Nota<on • There is a more appropriate way that we use in calculus to represent a func<on in wri<ng and that is: f(x) = y The input (variable) is listed in brackets Output Name of Func<on
- 7. Func<on Nota<on Examples Example: • f(x) = x + 1 • A(r) = πr2 • V(h,r) = πr2h
- 8. Rela<ons The topic of func<ons is in fact in sub-‐topic under him much broader subject In mathema<cs called Rela/ons. Defini<on: A rela<on is a set of ordered pairs. What it is an ordered pair? Well let us use this func<on as an example: f(x) = x2 + 1
- 9. Let us list our inputs from 1 to 5 and calculate their corresponding outputs: Now we can pair our inputs with our outputs in an orderly fashion like this:
- 10. Formal Defini<on of a Func<on • A func<on a special rela<on in which each element x in the Domain is paired using a rule, with exactly one and only one element f(x) in the Range. • There are two types of rela<ons that sa<sfy this criteria and they are called one-‐to-‐one and many-‐to-‐one rela<ons. • A one-‐to-‐many rela<on is NOT a func<on.
- 11. Example of One-‐To-‐One Func<on Consider the rela<on f: x → 2x + 5 given that the domain is Find the corresponding range values and hence draw a mapping diagram to represent the rela<on. 0, 1, 2, 3
- 12. One-‐To-‐One Func<ons Con<nued A func<on from set A to set B is said to be an One-‐To-‐One (injec<ve) func<on if no two or more elements of set A have the same elements mapped or imaged in set B.
- 13. Many-‐To-‐One Func<ons Consider the func<on f(x) = x2 and let our domain be {-‐2 ≤ x ≤ 2}.
- 14. Many-‐To-‐One Func<ons Con<nued A func<on from set A to set B is said to be a many-‐ to-‐one func<on if two or more elements in set A processed through the func<on produces the same output or same element in set B.
- 15. One-‐To-‐Many is Not A Func<on Consider the inverse of func<on f(x) = x2 in which we generate by exchanging the values for the domain and range. The inverse func<on follows the rule: f ’(x) = ± √x
- 16. A Visual Test For Func<ons We can use a very simple test called the Ver/cal Line Test to determine whether the Graph of A Rela/on in indeed a func<on or not. • Defini<on: – Given a curve drawn in the coordinate plane. Then this curve is a graph of a func<on if and only if no ver<cal line can be made to intersect the curve at more than one point.
- 17. Using The Ver<cal Line Test Consider the following graphs and decide which if any, are graphs of func<ons: