Relations & Functions

copyright © 2012 Lynda Aguirre2 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). DEFINITIONS 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 Aguirre3 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. 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 Aguirre4 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. x’s don’t repeat It IS a Function The y’s repeat, but that is not what we’re looking for xy xy xy x’s repeat It is NOT a Function x’s don’t repeat It IS a Function

Is This A Function? copyright (c) 2012 Lynda Aguirre5 The information is given in several different ways What to look for: If any of the x’s repeat, it is not a function. Two bubbles with arrows 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 Aguirre6 The information is given in several different ways For 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 Aguirre7 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 Problems Which of these are Functions? copyright (c) 2012 Lynda Aguirre8 xy xy FUNCTION NOT A FUNCTION FUNCTION

Function Notation copyright (c) 2012 Lynda Aguirre9

Function Notation copyright (c) 2012 Lynda Aguirre10 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 Aguirre11 Let and. Find the value of the following:

Function Notation copyright (c) 2012 Lynda Aguirre12 Let and. Find the value of the following: -2

Domain & Range copyright (c) 2012 Lynda Aguirre13

Domain & Range 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) Domain: -2, 1, 0 Range: 3,-2,5

Domain of a Square Root Function Square 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

Practice Problems Domain of a Square Root Function copyright (c) 2012 Lynda Aguirre16 Domain: [0, ∞)

Domain of a Rational Function Rational 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, ∞)

Practice Problems Domain of a Rational Function copyright (c) 2012 Lynda Aguirre18 Domain: (-∞, -1)U(-1, ∞) Domain: (-∞, 0)U(0, ∞)