a relation is a correspondence between two sets. if x and y are two elements in these sets and if a...
DESCRIPTION
Example: Given the relation {(-2,-3),(2,3),(-1,2),(-3,4),(3,4)} Is this relation a function?_______________ (does any one x-value have more than one corresponding y value?) What is the domain? -3,-2, -1, 2, 3 What is the range? -3, 2, 3, 4 Example: Given the relation {(-4,4),(-2,2),(0,0),(-2,-2)} Is this relation a function?_______________ (does any one x-value have more than one corresponding y value?) What is the domain?___________________ What is the range?______________________TRANSCRIPT
A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then x corresponds to y, or y depends on x.
The set of x-coordinates {2,3,4,4,5,6,6,7} corresponds to the set of y coordinates {60,70,70,80,85,85,95,90}
The set of distinct x-coordinates is called the domain of the relation. This is the set of all possible x values specified for a given relation.The set of all distinct y values corresponding to the x-coordinates is called the range.In the example above, Domain = {2,3,4,5,6,7} Range = {60,70,80,85,95,90}
Math 80A Section 3.5 Relations and Functions
Example:
xHours Studying in Math Lab 2 3 4 4 5 6 6 7
y Score on Math Test 60 70 70 80 85 85 95 90
A function, f, is like a machine that receives as input a number, x, from the domain, manipulates it, and outputs the value, y.
The function is simply the process that x goes through to become y. This “machine” has 2 restrictions:
1. It only accepts numbers from the domain of the function.2. For each input, there is exactly one output (which may be repeated for different inputs).
Input x
Output yFunction
f(x)
FUNCTIONS
“OFFICIAL” DEFINITION OF A FUNCTION:Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X, exactly one element of Y.However, an element of Y may have more than one elements of x associated with it.That is, for each ordered pair (x,y), there is exactly one y value for each x, but there may be multiple x values for each y. The variable x is called the independent variable (also sometimes called the argument of the function), and the variable y is called dependent variable (also sometimes called the image of the function.)
Analogy: In the x-y “relation”-ship, the x’s are the wives and the y’s are the husbands. A husband is allowed to have two or more wives, but each wife(x) is only allowed 1 husband(y).
This relation is not a function because there are two different y-coordinates for the x-coordinate, 4, and also for the x-coordinate, 6.
Example:Given the relation {(-2,-3),(2,3),(-1,2),(-3,4),(3,4)}Is this relation a function?_______________(does any one x-value have more than one corresponding y value?)
What is the domain? -3,-2, -1, 2, 3What is the range? -3, 2, 3, 4
Example:Given the relation {(-4,4),(-2,2),(0,0),(-2,-2)}Is this relation a function?_______________(does any one x-value have more than one corresponding y value?)
What is the domain?___________________What is the range?______________________
A relation is not a function of x if there is more than one corresponding y-value for any x-value.To be a function of x, put the relation in the form y = ___ and there should be only 1 possible solution for any one x-value.
Example 6 p. 202
Are these functions of x?
a) x = y2
y can be or
Not a function of x.
b) y = 2x There is only 1 possibility for any x.
So YES. it is a function of x.
c) x = |y|
Solve for y. x = y or x = -y
y = x or y = -x
Two possibilities, therefore NOT A FUNCTION.
xy
x x
VERTICAL LINE TEST
A graph is the graph of a funciton if and only if there is no vertical line that crosses the graph more than once.
ExampleConsider the equation for the line segment: y = 2x – 5, where the domain is {x|1 ≤ x ≤ 6}Is this equation a function?Notice that for any x, you can only get one answer for y. (E.g. when x =1, then y = 2(1) – 5= -3.) Therefore the equation is a function. Functional notation for this equation would bef(x) = 2x – 5Just replace the y with f(x). Note: This is stated “f of x”,It does not mean f times x, though it looks like that.What is the range?Since this is a straight line, we need only check y values at endpoints of domain. The y values do not fluctuate between these endpoints. The y values are also called function values, so they are often referred to as f(x), which means the value of the function at x (not f times x).The endpoints of the domain are 1 and 6.f(1) = the value of the function (what is y?) when x = 2 = 2(1) – 5 = -3f(6) = the value of the function (what is y?) when x = 6 = 2(6) – 5 = 7So the range is {y|-3 ≤ y ≤ 7}
ExampleFind the range of the function given by the equation f(x) = -3x + 2 if the domain is {-4,-2,0,2,4}.Recall domain means possible x-values.
The ordered pairs that belong to thisfunction are graphed to the right. They are:{(-4,14), (-2,8), (0,2),(2,-4),(4,-10)}Range means possible y values, so Range = {-10,-4,2,8,14}
-4
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-1 0 1 2 3 4 5 6 7
This figure is a line segment with endpoints(1,-3) and (6,7).
x f(x) = -3x + 2-4 14-2 80 22 -44 -10
-5 -4 -3 -2 -1 0 1 2 3 4 5
-16
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Series1; 14
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f(x) = -3x + 2
x-axis
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is