Post on 31-Dec-2015
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(2-1) Relations and Functions
Cartesian Coordinate Plane
Def:
Composed of the
x-axis (horizontal) and the y-axis (vertical) which meet at the origin (0,0).
Relation
Def: Set of ordered pairs
Example: {(3,2),(4,1),(7,3),
(1,2),(6,3),(5,6)}
Quadrants
Def:
The Cartesian
coordinate plane is divided into four quadrants (parts), which can be written in the form (x,y).
Domain
Def:
Domain of a relation is the set of all x-values in a set of ordered pairs.
Domain
Def:
Domain of a relation is the set of all x-values in a set of ordered pairs.
x,y x,y x,y x,y x,y x,y
Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}
Domain
Def:
Domain of a relation is the set of all x-values in a set of ordered pairs.
x,y x,y x,y x,y x,y x,y
Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}
Domain: { 3, 4, 7, 1, 6, 5 }
Range
Def:
The range of a relations is the set of all
y-values in a set of ordered pairs.
x,y x,y x,y x,y x,y x,y
Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}
Range: { 2, 1, 3, 2, 3, 6 }
Range: { 2, 1, 3, 6} , you don’t repeat
Mapping
Def:
Shows how the member of the domain is paired with each member of the range.
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X Y
f(x)
Example: {(3,2),(4,1),(7,3),(1,2),(6,3),(5,6)}
x,y x,y x,y x,y x,y x,y
Function
Def:
A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.
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X Y
f(x)
Example: {(2,4),(1,2),(5,4),(4,2)}
x,y x,y x,y x,y
Not a Function
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X Yf(x)
Example: {(2,1),(4,3),(2,4),(1,1)}
x,y x,y x,y x,y
This is not a functionbecause 2 goes to 1 and 3, which contradicts the definitionof a function.
Discrete Vs. Continuous
Discrete is a relation in which the Domain is a set of individual points.
Continuous is when the domain of a relation has an infinite number of elements and the relation can be graphed with a line or smooth curve.
Independent Variable
When an equation represents a function, the variable, usuallyx, whose values make up the domain is called the Independent Variable.
y = 3x+4
Independent Variable
Dependent Variable
The other variable, usually y, is called the Dependent Variablebecause its values depend on x.
y = 3x+4
Independent Variable
Dependent Variable
Vertical Line Test
If no vertical line intersects a graph in more than one point, the graph represents a function.
Vertical Line Test
If no vertical line intersects a graph in more than one point, the graph represents a function.
Thus this is not a function becausethe vertical line crosses the graph through more than one point.
Evaluating a Function
Given ,find each value.
(a) f (-3)
Take original function
Substitute
Simplify= 9+2=11
Evaluating a Function
Given ,find each value.
(b) f (3z)
Take original function
Substitute
Simplify
Class WorkDetermine whether each relation is a function. Write yes or no.
Discrete or ContinuousGraph each relation or equation and find the domain and rang. Next determineif the relation is discrete or continuous. Then determine whether the relation or equation is a function.
Evaluate