Lesson 1: Functions and Relations Flashcards
Set of x values
Domain
Set of y values
Range
Ordered pairs
(x,y)
A rule that relates the values from a set of values (called the domain) to a second set of values (called the range).
Relations
In relations, inputs generate:
one or more outputs
A relation where each element in the domain is related to only one value in the range by some rule
Function
In functions, inputs generate:
only one output
A function is a set of ordered pairs where:
no ordered pairs have the same x value
between two sets X and Y is simply a subset of the Cartesian product X x Y, i.e., a collection of ordered pairs (x,y) where x ∈ X and y ∈ Y
Relation
∈ means
“is an element of”
Let x and Y be sets. Then a function from X to Y, denoted by f: X -> Y, is a set f of ordered pairs in X x Y such that for each x ∈ X there exists a unique y ∈ Y with (x,y) ∈ f. In other words, if (x,y) ∈ f and (x,y’) ∈ f, then y=y’
Function
Function Notation
f(x)
f(x) is read as
“f of x” or “function of x”
in f(x)=y, what are the variables?
- x is the independent variable
- y is the dependent variable
A function can be illustrated as:
a machine where there is an input and output
input: person | output: blood type
Function
A person can only have 1 blood type
input: airport| output: airport code
Mere Relation
Airports have one code from the IATA and ICAO
input: resident | output: address
Function
A person can only have one address/live in one place
input: students | output: subject teachers
Mere Relation
A student can have more than one subject teacher.
input: items | output: expiration date
Mere Relation
Items can be made on different dates. thats why different exp. dates.
A function that is defined by two or more equations over a specified domain
piecewise-defined function
Piecewise functions can be denoted by:
formula 1 if x is in domain 1
f(x) = {formula 2 if x is in domain 2
formula 3 if x is in domain 3
A function that takes an input as a real number and gives an output that is the greatest integer less than or equal to the resulting number.
Floor Function
Floor functions are denoted by:
⌊ ⌋, [ ], or〚 〛
A function that takes an input as a real number and gives an output that is the least integer greater than or equal to the resulting number.
Ceiling Function
Ceiling functions are denoted by:
⌈ ⌉, ] [, or 〛〚
Key phrase of floor and ceiling functions
Hour (or a fraction of an hour)
⌈4.5⌉ =
5
⌈-4.5⌉ =
-4
⌊3.7⌋
3
⌊-3.7⌋
-4
means replacing the variable in the function with a value from the function’s domain and computing the result
Evaluating a function
In the function f(x), if we are evaluating f at x = a, we write:
f(a)
