Question

Which set of ordered pairs does not represent a function?

{(3, 7), (–1, 9), (–5, 11)}

{(9, –5), (4, –5), (–1, 7)}

{(–2, 1), (3, –4), (–2, –6)}

Answer 1

The last one

Explanation:

A function has to return a unique value when given an argument. In the last set `{(–2, 1), (3, –4), (–2, –6)}`, the argument -2 is supposed to return both 1 and -6 : this is not possible for a function.

Additional technical points

There is another important part of the definition of a function that we should really worry about here. A function is defined with a domain - the set of input values that it takes, as well as a codomain - the set of possible values it can return (some books call this range).

A function has to return a value for each element of the domain. Since the domain has not been specified for any of the prospective functions here, we can not be sure that even the other two fit the criteria to be a function.

What we can say is :

• `{(3, 7), (–1, 9), (–5, 11)}` can represent a function if the domain is specified as the set `{3,-1,-5}`

• `{(9, –5), (4, –5), (–1, 7)}` can represent a function if the domain is specified as the set `{9,4,-1}`

In both cases the codomain can be taken to be the set of integers (it is not demanded of a function that it returns every value in the codomain - just that every value it does return is in the codomain)

Answer 2

`" "`
`color(blue)("Set C"` does NOT represent a function.

Explanation:

`" "`
Given: Three Sets of Relations, say `color(red)(A, B,)` and `color(red)(C.`

Definition of a Relation:

A relation is simply a set of input and output values, represented in ordered pairs.

Any set of ordered pairs may be used in a relation.

No special rules are available to form a relation.

Definition of a Function:

A function is a set of ordered pairs in which each x-element has Only One y-element associated with it.

Examine the three sets of relations given to determine if any of them strictly follows the rule for being a function.

`color(green)("Step 1")`

Set the Input data table up:

`color(green)("Step 2")`

Rewrite the data table to facilitate comparing `color(red)(x` values of each set:

A simple visual examination tells us that `color(red)("Set C"` has `color(blue)(x = -2` twice.

Note that `color(red)("Set B"` uses the value `color(blue)((-5)` twice for the y-coordinate.

But, x-coordinate values are NOT repeated.

Set B is a function using the rule.

Hence,

`color(blue)("Set C"` does NOT represent a function.

`color(green)("Step 3")`

Plot ordered pairs of `color(blue)("Set A"` on a Cartesian coordinate plane:

`color(green)("Step 4")`

Plot ordered pairs of `color(blue)("Set B"` on a Cartesian coordinate plane:

`color(green)("Step 5")`

Plot ordered pairs of `color(blue)("Set C"` on a Cartesian coordinate plane:

`color(red)(C_1(-2,1), C_3(-2,-6)` have the same x-coordinate value.

Hope it helps.