# 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)}

### Comments for Function - three ordered pairs?

 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. ----------------------------------------------

 Nov 29, 2012 Function by: Staff ---------------------------------------------- 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 ----------------------------------------------

 Nov 29, 2012 Function by: Staff ----------------------------------------------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 ---------------------------------------------- Part 4 Thanks for writing. Staff www.solving-math-problems.com