MMW 5&6

Language of relations and functions

relation

a set of ordered pairs

tabular, mapping diagram, graphical

ways to represent relations

one-to-one

all x has a

one-to-many

all x has a corresponding y, some x corresponds to more than one y

many-to-one

more than one x corresponds to one y

many-to-many

more than one x corresponds to one y (vice versa)

R = (x, y) E A x B

equation form of relation of sets

domain of R

• set of all the possible values for x (dom R)

• set A is called _____ if x R y, if and only if (x, y) is in R

co-domain

• set B is called _____ if x R y, if and only if (x, y) is in R

• im R

range

set of all the possible values for y (im R)

functions

relation with domain A and co-domain B that satisfies the ff:

• for every x in A, there is a corresponding y in B such that (x, y) E F

• for all elements x in A, and y & z in B, (x, y) E F & (x, z) E F, y = z

• “for every x corresponds a unique value of y”

• y = f(x)

one-to-one, many-to-one

relations of x and y that can be used in functions for a function is a relation that for each input, there is only one output

dependent variable

y is the… (output)

name of function

f is the…

independent variable

x is the… (input)

constant function

• f(x) = c (y = c)

• The c-value can be any number

• so the graph is a horizontal line

• domain = R, range = {y}

domain = R

write “ domain = the set of all real numbers”

vertical line test

how can you determine if a relation is a function?

identity function

• f(x) = x (y = x)

• the x-value is the same as the y-value

• graph  is a diagonal line going through the origin

• domain = R, range = R

linear function

• f(x) = mx + b (y = mx + b)

• an equation written in slope-interception form

• graph is a straight line

• domain = R, range = R

absolute value function

• f(x) = |x| (positive values)

• V-shaped graph

• domain = R, range = R >= 0

rational function

• f(x) = n(x) / d(x)

• both numerator and denominator are functions and d(x) ≠ 0 

• graph is two curving lines getting close to each other but never meeting

binary operation

calculation that combines two elements to produce another element