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Function - three ordered pairs?











































- Define a mathematical relation and function.

- Is the following set of three ordered pairs a function?

          {(3,4), (4,4), (5,4)}

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Nov 29, 2012
Function
by: Staff



Answer



Part 1

Define a RELATION


A “relation” is a mathematical statement which shows how one quantity is related to at least one other quantity.

A relation can be an equation, a diagram, or simply a list which shows how one set of elements is related to another set of elements.

A “binary” relation is a set of ordered pairs. For example {(1,1), (1,2), (3,4), (9,9)}.

A "ternary" relation is a set showing the relationship between three elements (an ordered triple). For example {(1,1,9), (1,2,3), (3,4,5), (9,9,8)}.

The set of three ordered pairs in your problem statement is a binary relation: r.

          r: {(3,4), (4,4), (5,4)}



Define a FUNCTION


A function is a relation with certain qualifications.


(1) There can be ANY NUMBER of INDEPENDENT VARIABLES (these are the input variables: x, z, q, r, etc.)

The following example is a function with two independent (or input) variables

          Total Pay = (rate of pay) * (number of hours worked)

          In this example, “Total Pay” is the dependent variable. “Rate of pay”
          & “number of hours worked” are the two independent (or input) variables.


Normally, this relationship would be abbreviated:

          Total Pay = (rate of pay) * (number of hours worked)

          TP = p * h

Written in functional notation, it would be:

          f(p,h) = p*h


(2) There can be ONLY ONE DEPENDENT VARIABLE.

Using the same example introduced in (1):

          Total Pay = (rate of pay) * (number of hours worked)

          TP = p * h

          f(p,h) = p*h

          In this example, f(p,h) is the only dependent variable.


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Nov 29, 2012
Function
by: Staff


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Part 2

(3) The DEPENDENT VARIABLE can only be EQUAL TO A SINGLE VALUE AT A TIME. In other words, for every input value (or every combination of input values) there can only be a single output value.

Again, using the same illustration introduced in (1):

          Total Pay = (rate of pay) * (number of hours worked)

          TP = p * h

          f(p,h) = p*h

For every unique combination of “Rate of pay” & “number of hours worked” there is ONLY ONE VALUE OF “Total Pay”



EQUATIONS VS. FUNCTIONS


An equation and a function are not the same, although an equation can often be rewritten as a function.

An equation is always equal to a specific value on each side of the equal sign. Here are some examples of equations:

          x + 5 = 9x

          y + z = 130

          z³ + z² = 0



Only the second equation listed in the three examples can be rewritten as a function

          x + z = 130 is an equation

          y = 130 - z is a function

          f(x) = 130 - z is the same function, written in functional notation



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Nov 29, 2012
Function
by: Staff

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Part 3

ANSWER TO YOUR ORIGINAL QUESTION: Is the following set of three ordered pairs a function?


          {(3,4), (4,4), (5,4)}

(1) There can be ANY NUMBER of INDEPENDENT VARIABLES (these are the input variables: x, z, q, r, etc.)

          Is there at least one INDEPENDENT VARIABLE?

          YES.

          The ordered pairs have the form (input, output), usually expressed as (x, y).

          The independent variable is x (the first number in each pair).

          x = {3, 4, 5}

(2) Is there ONLY ONE DEPENDENT VARIABLE?

          YES.

          The ordered pairs have the form (input, output), which can be expressed as (x, y).

          There is only one dependent variable, y (the second number in each pair).

          y = {4}



(3) Is the DEPENDENT VARIABLE only EQUAL TO A SINGLE VALUE AT A TIME. In other words, for every input value, is there only a single output value?


          YES.

          For every input value of x, there is only a single value of y.

           (The fact that the output value of y is always equal to 4 is irrelevant to the definition of a function.)


Final Answer:

The set of three ordered pairs {(3,4), (4,4), (5,4)} represents a function.









Dec 10, 2012
Function
by: Staff


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Part 4

Math – The set of three ordered pairs {(3,4), (4,4), (5,4)} represents a function.








Thanks for writing.

Staff
www.solving-math-problems.com



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