relations & functions. copyright © 2012 lynda aguirre2 a relation is any set of ordered pairs....
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Relations & Functions
Relations & Functions
copyright © 2012 Lynda Aguirre 2
A RELATION is any set of ordered pairs.
A FUNCTION is a special type of relation where each value in the domain corresponds to a unique element of the range (i.e. the x-
values don’t repeat).
DEFINITIONSDEFINITIONS
The DOMAIN is a list of first coordinates in an ordered pair (x-values)
The RANGE is a list of second coordinates in an ordered pair (y-values)
Is This A Function?
copyright (c) 2012 Lynda Aguirre 3
The information is given in several different ways
A set of (x,y) coordinates
What to look for: If any of the x’s repeat, it is not a function.What to look for: If any of the x’s repeat, it is not a function.
x’s don’t repeat—It IS a Function
x’s repeat—It is NOT a Function
The y’s repeat, but that is not what we’re
looking for
x’s don’t repeat—It IS a Function
Is This A Function?
copyright (c) 2012 Lynda Aguirre 4
The information is given in several different ways
A table
What to look for: If any of the x’s repeat, it is not a function.What to look for: If any of the x’s repeat, it is not a function.
x’s don’t repeatIt IS a Function
The y’s repeat, but that is not what we’re
looking for
x y
3 2
0 -1
-1 4
x y
-1 5
2 -1
-1 4
x y
3 4
0 -1
-1 4
x’s repeatIt is NOT a Function
x’s don’t repeatIt IS a Function
Is This A Function?
copyright (c) 2012 Lynda Aguirre 5
The information is given in several different ways
What to look for: If any of the x’s repeat, it is not a function.What to look for: If any of the x’s repeat, it is not a function.
Two bubbles with arrows
x’s don’t repeatIt IS a Function
x’s repeatIt is NOT a Function The y’s repeat, but
that is not what we’re looking for
x’s don’t repeatIt IS a Function
Is This A Function?
copyright (c) 2012 Lynda Aguirre 6
The information is given in several different ways
For a figure, use the vertical line testFor a figure, use the vertical line test
THE VERTICAL LINE TEST: If you are given a graph, draw a vertical line through the figure, if it crosses any part of the figure more than once, it is not a function.
It IS a Function
It is NOT a Function
It IS a Function
EQUATIONS
copyright (c) 2012 Lynda Aguirre 7
If you are given an Equation, look at the power of the y-variable, if it is odd, the figure is a function, if the power of the y is even, the figure is
not a function.
It IS a Function It is NOT a Function
Practice ProblemsWhich of these are Functions?
copyright (c) 2012 Lynda Aguirre 8
x y
2 -7
5 -1
-1 0
x y
-1 7
0 -1
-1 -5
FUNCTIONFUNCTION
FUNCTIONFUNCTION
FUNCTIONFUNCTION
NOT A FUNCTIONNOT A FUNCTION
NOT A FUNCTIONNOT A FUNCTION
NOT A FUNCTIONNOT A FUNCTION
NOT A FUNCTIONNOT A FUNCTION
NOT A FUNCTIONNOT A FUNCTIONFUNCTIONFUNCTION
Function Notation
copyright (c) 2012 Lynda Aguirre 9
Function Notation
copyright (c) 2012 Lynda Aguirre 10
Old notation:
New notation:
These both mean the same thing: Plug in (-3) for “x” and solve
This represents the point (-3, -17)
The x-value is what you plugged in
The y-value is what it kicked out
Function Notation
copyright (c) 2012 Lynda Aguirre 11
Let and . Find the value of the following:
Function Notation
copyright (c) 2012 Lynda Aguirre 12
Let and . Find the value of the following:
-2
Domain & Range
copyright (c) 2012 Lynda Aguirre 13
Domain & RangeThe DOMAIN is a list of first coordinates in an ordered pair (x-values)
The RANGE is a list of second coordinates in an ordered pair (y-values)
Domain: -2, 1, 0
Range: 3,-2,5
Domain of a Square Root FunctionSquare roots (and other even roots) have restricted domains that can be
calculated by setting the radicand (value under the radical) ≥ zero and then solving the inequality.
Examples:
Domain: [0, ∞)
Domain: [-3, ∞)
Rule: When you multiply or divide an inequality by a negative number, the
inequality reverses direction
Rule: When you multiply or divide an inequality by a negative number, the
inequality reverses direction
Practice ProblemsDomain of a Square Root Function
copyright (c) 2012 Lynda Aguirre 16
Domain: [0, ∞)
Domain of a Rational FunctionRational Functions (Fractions) have restricted domains that can be
calculated by setting the radicand (value under the radical) ≠ zero and then solving the inequality.
Examples:
Domain: (-∞, 0)U(0, ∞)
Domain: (-∞, -3)U(-3, ∞)
Domain: (-∞, -3)U(-3, ∞)
Practice ProblemsDomain of a Rational Function
copyright (c) 2012 Lynda Aguirre 18
Domain: (-∞, -1)U(-1, ∞)
Domain: (-∞, 0)U(0, ∞)