3. Functions

Utah Math 2 - 2020 Edition

3.01 Review: Functions and relations

Lesson

A relation is a relationship between sets of information. For example, think of the names of the people in your class and their heights. If I gave you a height (e.g. $162$162 cm), you could tell me all the names of the people who are this tall and there may be more than one person. Let's say someone came to your class looking for the person who was $162$162 cm tall, that description might fit four people! There's not one clear answer. This data could be expressed as a relation.

A function is a special type of relation, where each input only has **one** output. We've already learned about the dependent and independent variables in equations. Functions work in a similar way, where the way the dependent variable, $f(x)$`f`(`x`), varies depending on the rule that is applied to the independent variable, $x$`x`. For example, if we think of a vending machine selling juice, the amount of money we have to pay (the dependent variable) depends on the amount of juice we want to buy (the independent variable). Let's say each bottle of juice cost $\$3$$3. If we bought one bottle, it would cost $\$3$$3, if we bought two bottles, it would cost $\$6$$6 and so on. Do you notice how the value of our independent variable (the number of bottles of juice) always produces a different, unique dependent variable (cost)? This is an example of a function.

Let's look at another example. Say we have the expression $f(x)=2x$`f`(`x`)=2`x`. Let's construct a table of values to record the results:

$x$x |
$-1$−1 | $0$0 | $1$1 | $2$2 |
---|---|---|---|---|

$f(x)$f(x) |
$-2$−2 | $0$0 | $2$2 | $4$4 |

See how each $x$`x` value gives a unique $f(x)$`f`(`x`) value? This means this data displays a function.

If you can draw a vertical line anywhere on a graph so that it crosses the graph in more than one place, then the relation is **not **a function.

Here is an example of a relation that is not a function. See how when I drew in the blue vertical line, it crossed the graph in two places?

In other words, functions have to pass the vertical line test **at every point. **

Here is one example of a function.

Here is another function.

If you can write a relationship between $x$`x` and $y$`y` then we can see that there is a relation. However, if this relationship only yields one value of $y$`y` for each $x$`x` value, then it is a function.

When we solve for $y$`y` in $2y-4x=10$2`y`−4`x`=10, we can tell whether it's a function or a relation.

$2y-4x=10$2`y`−4`x`=10

$2y=4x+10$2`y`=4`x`+10

$y=2x+5$`y`=2`x`+5

See that each value of $x$`x` only yields one $y$`y` value, which means that it is a **function**.

Let's try this process for the equation $y^2=x$`y`2=`x`:

$y=\pm\sqrt{x}$`y`=±√`x`

See how $y$`y` could be $\sqrt{x}$√`x` or $-\sqrt{x}$−√`x`? Since there are two possible values of $y$`y`, we can only say it is a **relation**.

Remember!

While all functions are relations, not all relations are functions.

Do the following sets of points describe a function or a relation?

**A)** $\left(1,5\right)$(1,5), $\left(1,1\right)$(1,1), $\left(7,-2\right)$(7,−2), $\left(-5,-10\right)$(−5,−10)

**Think:** Does each $x$`x` value have a unique $y$`y` value?

**Do:** There are $2$2 possible $y$`y` values when $x=1$`x`=1.

**Reflect**: This describes a relation.

**B)** $\left(1,5\right)$(1,5), $\left(7,-2\right)$(7,−2), $\left(-5,-10\right)$(−5,−10), $\left(13,-13\right)$(13,−13)

**Think:** Does each $x$`x` value have a unique $y$`y` value?

**Do:** In this set of coordinates, each $x$`x` value has only one unique $y$`y` value.

**Reflect**: This describes a function.

Determine whether the following graph describes a function or a relation.

**Think:** This graph passes the vertical line test. Therefore, it is a function.

Determine whether the following graphs describe relations, and whether they describe functions.

- Loading Graph...
Select all answers that apply.

Function

ARelation

BFunction

ARelation

B - Loading Graph...
Select all answers that apply.

Function

ARelation

BFunction

ARelation

B

The pairs of values in the table represent a relation between $x$`x` and $y$`y`. Do they represent a function?

$x$x |
$-8$−8 | $-7$−7 | $-6$−6 | $-3$−3 | $2$2 | $7$7 | $9$9 | $9$9 | $10$10 |
---|---|---|---|---|---|---|---|---|---|

$y$y |
$8$8 | $13$13 | $-18$−18 | $-16$−16 | $-15$−15 | $-2$−2 | $-4$−4 | $11$11 | $-9$−9 |

Yes

ANo

BYes

ANo

B

Oprah makes scarfs to sell at the market. It costs her $\$2$$2 to produce each one, and she sells them for $\$5$$5.

Consider when $1$1, $2$2, $3$3, $4$4 and $5$5 scarfs are sold. Graph the points representing the relation between the number of scarfs she manages to sell and her total profit.

Loading Graph...Is this relation a function?

No

AYes

BNo

AYes

B