A relation is a rule that relates an element of one set to one or more elements in another set
In a function, the input has exactly one output, creating a many-to-one correspondence where many inputs are mapped to one element
In an ordered pair x, y, x is the input and y is the output generated by a machine depending on some rule
In a relation, elements from set A are mapped to elements in set B using some rule f
Before starting this lesson, it is suggested to revisit the following topics from previous math lessons:
A relation is a set of ordered pairs x, y where x is an element of the domain of the relation, and y is an element of the range of the relation, related by some rule
An input can have more than one output, or different inputs can produce the same output
A function is a relation that relates an element of one set to exactly one element in another set by some rule
One-to-one correspondenceāØ
Each element in S is mapped to exactly one element in T
One-to-many correspondence
One input is mapped to multiple outputs
For a relation to be considered as a function, each input (x) should have exactly one output (y)
We defined a relation earlier as a set of ordered pairs x, y. How do we know whether it is a function or not?
Consider the relation f = 1,2 , 3,7 , 5,12 , 7,17. The relation f is a function as each x-value has exactly one y-value related to it
The relation in Figure 1.1.3 is not considered as a function. The inputs 1 and 3 have more than one output
Many-to-one correspondenceāØ
Many inputs are mapped to one element
A set of ordered pairs x, y can be considered a function if there is no repetition in the x-values, implying each x-value is related to only one y-value
Function since each input has exactly one output
Example 1: Determine if the following relations are functions
Solution: a. E = 0,1 , 1,4 , 2,7 , 3,10 The relation E is a function since no x-values are repeated implying that each x-value is related to only one y-value
The relation in Figure 1.1.2 is considered to be a function since each element in S is mapped to exactly one element in T
Consider another relation g = 2, -3 , 0,2 , -2, -1 , 2,3. The relation g is not a function as x = 2 is related to two y-values, -3 and 3
RelationsāØ
E = 0,1 , 1,4 , 2,7 , 3,10
F = 2,4 , -3,9 , 0,0 , -1,1
G = C D , 3 , -2,0 , 0,4 , 1, -6 , -2,8
A relation is considered as a function if it is one-to-one or many-to-one
Relation is a function if each š„-value is related to only one š¦-value
Relation š¹ is a function since no š„-values are repeated, implying that each š„-value is related to only one š¦-value
Definition of Vertical Line Test: If a vertical line intersects the graph of a relation at exactly one point, then the graph represents a function. Otherwise, the graph is not a function
The graph of a relation where a vertical line intersects at two points implies that the š„-value has two š¦-values related to it, making it not a function
Relation šŗ is not a function since š„ = 2 is repeated in the relation, implying that this š„-value is related to more than one š¦-value; thus, violating the definition of a function
Equations can describe functions and relations
The graph that intersects the vertical line test at exactly one point represents a function
The equation šš ā š = š is not specified as a function or not
The equation š¦ = 4 ā š„D is not a function as it generates two š¦-values for ļæ½ļæ½ = 0
If there exists an š„-value that generates more than one š¦-value, then the equation is not a function
The relation š¦ = š„F + 2 is considered a function as each š„-value has only one š¦-value
The vertical line test can be drawn anywhere that will intersect the graph
Functions can represent real-life situations
The equation š„ + š¦ = 1 is a function as no x-values generate more than one y-value