Essential Question: What separates a relation from a function?

A relation is a set of pairs of input and output values. You can write a relation as a set of ordered pairs. Input (time): {0 0.1 0.2 0.3 0.4}

relation: {(0, 10), (0.1, 9.8), (0.2, 9.4), (0.3, 8.6), (0.4, 7.4)}

Output (height): 0, {10 0.1, 9.8 0.2, 9.4 0.3, 8.6 0.4, 7.4}

The 1st input goes with the 1st output, 2nd input with 2nd output, 3rd input with 3rd output, etc.

You can graph a relation on a coordinate plane. Example 1: Graph the relation

{ , , , } The first number represents

the “x” (left/right) The second number

represents the “y”(up/down)

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y(-2, 4) (3, -2) (-1, 0) (1, 5)

Your Turn Graph the relation:

{(0, 4), (-2, 3), (-1, 3), (-2, 2), (1,-3)}

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

The domain of a relation is the set of all inputs (x-coordinates)

The range of a relation is the set of all outputs (y-coordinates) The way to keep all that straight?▪ “d” comes before “r”▪ “x” comes before “y”▪ So, in an ordered pair, the first number is the

domain & x-coordinate. The second number is the range & y-coordinate

Example 2: Finding domain and range from a graph. Find the domain and range from the

relation below.

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Relation:{(-3, 2), (0, 1), (2, 4), (4, -3)}

Domain:{-3, 0, 2, 4}

Range:{-3, 1, 2, 4}

Your Turn Find the domain and range from the

relation below.

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y Relation:{(-3, 1), (-1, 1), (1, 1), (1, 3), (-1, -2), (-1, -4), (1, -4)}

Domain:{-3, -1, 1}

Range:{-4, -2, 1, 3}

Another way to show a relation is by a mapping diagram. A mapping diagram places the domain and range in boxes, and draws arrows to connecting elements.

Example 3: Make a mapping diagram for the relation {(-1, -2), (3, 6), (-5, -10), (3, 2)}

-5

-1

3

-10

-2

2

6

Domain Range

Your Turn: Make a mapping diagram for the relation {(2, 8), (-1, 5), (0, 8), (-1, 3), (-2, 3)}

-2

-1

0

2

3

5

8

Domain Range

A function is a relation where each element of the domain is paired to exactly one element in the range Meaning: No element of the domain gets repeated Example 4: Using mapping diagrams

-2

0

5

-1

3

4

Domain Range

-1

0

2

3

-1

3

5

Domain Range

Not a function Is a function

Your Turn: Which of the following are functions?

-1

0

1

-3

7

10

Domain Range

2

3

4

7

5

6

8

Domain Range

Not a functionIs a function

If we’re given a graph, we can use the vertical line test to determine whether a relation is a function We make a vertical (up/down) line with some straight object

(ruler, pencil), and move it from left to right. If the graph ever touches our line more than once, it is not a

function

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Is a functionNot a function

Your Turn: Which of the following are functions?

Is a function Not a function

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

A function rule expresses an output value in terms of an input value. y = 2x f(x) = x + 5 C = πd

output input You read function notation f(x) as “f of x” or

“a function of x”. Note that it doesn’t mean “f times x” Whenever you get a value in the parenthesis, it means

you substitute that value for x in the function. Example: f(3) = 3 + 5 = 8

Find f(-3), f(0) and f(5) for each function. Example 6: f(x) = 3x – 5▪ f(-3) = 3(-3) – 5 = -9 – 5 = -14▪ f(0) = 3(0) – 5 = 0 – 5 = -5▪ f(5) = 3(5) – 5 = 15 – 5 = 10

Your turn: f(a) = ¾a – 1▪ f(-3) = ▪ f(0) = ▪ f(5) =

¾(-3) – 1 = -9/4 – 1 = -13/4

¾(0) – 1 = 0 – 1 = -1

¾(5) – 1 = 15/4 – 1 = 11/4

Assignment Page 59 Problems 1 – 29 (odd problems)