3. Algebraic Relations & Functions

Lesson

We have explored relations and functions as ways of linking two sets of data, usually written using $x$`x`'s and $y$`y`'s as coordinates. We also learned that relations can only be functions if there's only one $y$`y` value associated with each $x$`x` value.

For example, $\left\{\left(1,2\right),\left(5,3\right),\left(2,-7\right),\left(5,-1\right)\right\}${(1,2),(5,3),(2,−7),(5,−1)} is not a function because the same $x$`x` value of $5$5 can be related to the $y$`y` values of both $3$3 and $-1$−1. We also know that graphically, we can use the vertical line test to see if a relation is a function.

Careful!

A relation where a $y$`y` value is associated with multiple $x$`x` values can still be a function! Do not confuse this with the criteria for a function where every $x$`x` value must be associated with only one $y$`y` value.

Domain and range both describe the span of values that a relation can take. Their definitions are very similar, but the small difference is very important.

Remember!

Domain - all of the possible $x$`x` values of a relation

Range - all of the possible $y$`y` values of a relation

We can find the domain and range of a relation no matter how it is represented. We simply need to look at the ordered pairs, graph, table, or other representation and list out all of the possible $x$`x` and $y$`y` values that the relation can have, putting commas in between each value and curly braces on the outside. Do not repeat any values that show up more than once. Typically the domain and range are written in ascending order (least to greatest) but that is not a requirement.

Let's look at a set of ordered pairs first.

Consider the relation $\left\{\left(1,2\right),\left(5,3\right),\left(2,-7\right),\left(5,-1\right)\right\}${(1,2),(5,3),(2,−7),(5,−1)}. State the domain and range.

**Think:** First look at the $x$`x`-coordinates to determine the domain, then look at the $y$`y`-coordinates for the range.

**Do:** List out the values for the domain and range. The domain is $\left\{1,2,5\right\}${1,2,5} and the range is $\left\{-7,-1,2,3\right\}${−7,−1,2,3}.

**Reflect:** Notice the domain and range are written in ascending order (least to greatest). This helps keep things organized but it is not a requirement. Also notice that the value of $5$5 was only included once in the domain. That is because we are only concerned with what all of the possible $x$`x` values are and not how many times they showed up.

A cafe is running a promotion where every $3$3 cups of coffee earns you a free cup, and each cup costs $\$3.50$$3.50. The table below lists the prices for $1$1 to $8$8 cups of coffee. State the domain and range.

Number of cups |
Price ($) |
---|---|

$1$1 | $3.50$3.50 |

$2$2 | $7$7 |

$3$3 | $10.50$10.50 |

$4$4 | $10.50$10.50 |

$5$5 | $14$14 |

$6$6 | $17.50$17.50 |

$7$7 | $21$21 |

$8$8 | $21$21 |

**Think:** Let the number of cups represent the $x$`x` values and the price represent the $y$`y` values.

**Do:** For the domain list out the $x$`x` values and for the range list out the $y$`y` values. The domain is $\left\{1,2,3,4,5,6,7,8\right\}${1,2,3,4,5,6,7,8} and the range is $\left\{3.50,7,10.50,14,17.50,21\right\}${3.50,7,10.50,14,17.50,21}. Notice that $10.50$10.50 and $21$21 were only included once.

**Reflect: **Consider what the domain and range represent in this context. The domain represents the possible number of cups of coffee someone could buy and the range represents the amount of money they could possibly have to pay for the coffee. If a person kept going back and buying coffee then we could keep adding to the domain and range!

Does this table represent a function? Remember that a relation is a function if each $x$`x` value is associated with only one $y$`y` value. It is easy to want to say that this is not a function because some of the $y$`y` values are repeated. But each number of cups ($x$`x`) is associated with one price ($y$`y`). So yes, this is a function!

State the domain and range of the graph below.

**Think:** It may be helpful to list out the ordered pair of each point to help identify all $x$`x` and $y$`y` values.

**Do:** The points on the graph are $\left\{\left(-1,2\right),\left(0,0\right),\left(1,2\right),\left(2,4\right)\right\}${(−1,2),(0,0),(1,2),(2,4)}. So the domain is $\left\{-1,0,1,2\right\}${−1,0,1,2} and the range is $\left\{0,2,4\right\}${0,2,4}.

Consider the relation in the table.

$x$x |
$y$y |
---|---|

$1$1 | $3$3 |

$6$6 | $2$2 |

$3$3 | $7$7 |

$8$8 | $1$1 |

$2$2 | $2$2 |

What is the domain of the relation? Enter the values, separated by commas.

What is the range of the relation? Enter the values separated by commas.

Is this relation a function?

Yes

ANo

BYes

ANo

B

Consider the relation on the graph below.

Loading Graph...

What is the domain of the relation?

Enter each value on the same line, separated by commas.

What is the range of the relation?

Enter each value on the same line, separated by commas.

Is this relation a function?

Yes

ANo

BYes

ANo

B

Determine the domain and range of a function