2.3 RELATIONS AND FUNCTIONS

CONSIDER y = 2x + 5 One way to display solutions is through a MAPPING DIAGRAM

Any set of ordered pairs is called a RELATION. RELATIONS Any set of ordered pairs is called a RELATION. Ex: the set of points { (1, 0), (-2, 3), (4, -1), (9, 3) } is a relation.

BUT…a line is a set of many points…. RELATIONS We can use a graph to picture a relation. BUT…a line is a set of many points….

…so, any equation y=mx+b is a relation. RELATIONS …so, any equation y=mx+b is a relation.

What is the domain and range for the set of points The list of x-values used in a relation is called the DOMAIN. The list of y-values used in a relation is called the RANGE. What is the domain and range for the set of points { (1, 0), (-2, 3), (4, -1), (9, 3) }?

For { (1, 0), (-2, 3), (4, -1), (9, 3) } DOMAIN AND RANGE We say the domain is D = { 1, -2, 4, 9 }. We say the range is R = { 0, 3, -1 }

CONSIDER x = y2

FUNCTIONS A FUNCTION is a relationship between two variables such that each value of the first variable is paired with exactly one value of the second variable.

CONSIDER x = y2 Therefore, x = y2 is NOT a function!

WHY??? FUNCTIONS { (1, 0), (-2, 3), (4, -1), (9, 3) } { (1, 0), (-2, 3), (1, -1), (9, 3) } IS a function!!! IS NOT a function!!! WHY???

FUNCTIONS AND GRAPHS Is y = 2x + 1 a function? List some ordered pairs… ( -1, -1 ) ( 4, 9 ) ( 1, 3 ) ( 5, 11 ) ( -2, -3 ) ( 0, 1 ) ( 3, 7 ) Could we ever re-use an x?

Any equation y = mx + b is a function. FUNCTIONS AND GRAPHS Since no x is repeated, y = 2x +1 IS a function!!! Any equation y = mx + b is a function.

FUNCTIONS AND GRAPHS Does this graph represent a function? HINT: What are coordinates of the marked points?

FUNCTIONS AND GRAPHS The circle is NOT a function! The marked points are ( 0, 2 ) and ( 0, -2 ) The circle is NOT a function!

FUNCTIONS AND GRAPHS Look at these graphs. Can you tell which are functions?

FUNCTIONS AND GRAPHS The vertical line test: If any vertical line intersects a graph at more than one point, the graph is NOT a function.

FUNCTIONS AND GRAPHS This graph fails the VLT, and is NOT a function!

FUNCTIONS AND GRAPHS This graph passes the VLT, and IS a function!

FUNCTIONS AND GRAPHS This graph fails the VLT, and is NOT a function!

FUNCTION NOTATION The height of an object x seconds after launch can be expressed by the function: HOW TO SAY IT….”f of x” x is the INDEPENDENT variable (input) f(x) is the DEPENDENT variable (output)

FUNCTION NOTATION What is the height of the object 3 seconds after launch? What is the height of the object 4.5 seconds after launch?

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