General Mathematics Lecture #1: Relations & Functions

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Questions and Answers

What type of correspondence represents a function based on the definition provided?

One-to-Many Correspondence

Many-to-Many Correspondence

Many-to-One Correspondence (correct)

One-to-One Correspondence (correct)

Which of the following makes a relation NOT a function?

If the relation is expressed verbatim

If the first elements are distinct

If the relation is defined by a graph

If two distinct members share the same first element (correct)

Which method determines if a graph represents a function?

Average Value Test

Horizontal Line Test

Vertical Line Test (correct)

Correspondence Mapping Test

Based on the rules of correspondence, when is a relationship functionally defined?

If the relationship indicates distinct values Signup and view all the answers

Given the set A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}, is A a function?

No, there are repeating first elements Signup and view all the answers

In the context of equations, what condition must the exponent of the y variable satisfy for it to represent a function?

It cannot be greater than one Signup and view all the answers

Which of the following pairs represents a many-to-one correspondence?

(1, 2), (2, 2), (3, 2) Signup and view all the answers

What does a rule of correspondence analyze?

The plurals and singular form in descriptions Signup and view all the answers

What is the domain of the relation A = {(−3, −4), (−2, 5), (−1, 5), (0, 6)}?

{−3, −2, −1, 0} Signup and view all the answers

Which of the following defines a function?

A set of ordered pairs with distinct first elements. Signup and view all the answers

What is the range of the relation B = {(−2, 5), (0, 5), (2, 5), (4, 5)}?

{5} Signup and view all the answers

Which of the following relations represents a function?

{(5, 6), (6, 7), (7, 8)} Signup and view all the answers

In the relation C = {(0, 3), (0, 6), (−2, −4), (−4, −5)}, what is the domain?

{0, −2, −4} Signup and view all the answers

Which statement is true regarding the vertical line test?

A function passes the vertical line test if a vertical line intersects the graph at exactly one point. Signup and view all the answers

For the relation D = {(1, 0), (0, 1), (−1, 0), (0, −1)}, what is the range?

{0, 1, −1} Signup and view all the answers

Which of the following ordered pairs confirms that a set is not a function?

(2, 3), (2, 5) Signup and view all the answers

Study Notes

Key Concepts in Relations & Functions

A relation is defined as a set of ordered pairs (x, y).
Example of a relation: A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
Ordered pairs are also known as coordinates, where:
- Abscissa refers to the x-value.
- Ordinate refers to the y-value.

Domain and Range

Domain: The set of all x-values in a relation.
- Example: Domain of set A = {1, 2, 3, 4}.
Range: The set of all y-values in a relation.
- Example: Range of set A = {3, 5, 9, 11, −15}.

Determining Domain and Range

Various relations provided for practice:
- A = {(−3, −4), (−2, 5), (−1, 5), (0, 6)}.
- B = {(−2, 5), (0, 5), (2, 5), (4, 5)}.
- C = {(0, 3), (0, 6), (−2, −4), (−4, −5)}.
- D = {(1, 0), (0, 1), (−1, 0), (0, −1)}.
- E = {(-4, 2), (-1, 3), (-4, 5), (-1, 0)}.

Understanding Functions

A function is a special type of relation where no two distinct ordered pairs have the same first element (x-value).
All functions are relations, but not all relations qualify as functions.

Ways to Represent a Function

Numerically:
- No shared x-coordinates among ordered pairs.
- Representation through ordered pairs or a table of values.
Graphically:
- Use the vertical line test; a function's graph will intersect any vertical line at exactly one point.
Mapping:
- Types include one-to-one, one-to-many, many-to-one, and many-to-many correspondences.
Algebraically:
- The exponent of the y variable must not exceed one.
- Example: Equation representation, such as 2x − 3y = 6.
Verbally:
- Formulated rules of correspondence expressed in words (like the relation between height and shoe size).

Function Verification

A relation is a function if:
- No two distinct pairs share the same x-coordinate.
- A vertical line intersects the graph at one point.
- In mappings, only one-to-one and many-to-one are considered functions.
- Algebraic representation satisfies the degree condition of y.
- Rules of correspondence indicate singularity in terms.

Classification of Relations and Functions

Analyze given sets and equations to deem whether they are relations or functions.
Examples provided for classification include:
- A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
- Various mathematical relations and their contextual interpretations such as plate numbers or student references.
Apply criteria to equations like y = √(x² − 1) or polynomials to assess if they define functions.

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Description

This quiz covers the fundamentals of relations and functions, focusing on defining key terms such as relation, domain, and range. Participants will determine the domain and range of various relations and assess whether given sets represent functions. It's an essential starting point for understanding mathematical relationships.