**A function is a set of ordered pairs where no two first values are the same. **

To determine whether a set of ordered pairs is a function or a relation:

- Look at the first numbers in the ordered pairs
- If all of the first numbers are different, then the set of ordered pairs is a function
- If any two first numbers are the same, then the set of ordered pairs is a relation

**Examples: Identify each set of ordered pairs as a relation or a function. List the domain and range.**

- {(-4,3),(5,6),(-4,8),(5,-3)}

Relation

Domain: {-4,5}

Range: {-3,3,6,8}

Look at the first numbers or the

x-values in the ordered pairs

Since there are two -4s, and two 5s, the set of ordered pairs is a relation

List the domain or x-values

as {-4,5} because we list each number just once and in ascending order

List the range or y-values

as {-3,3,6,8} because we list the numbers in ascending order - {(-2,6),(-1,5),(0,6),(1,5)}

Function

Domain: {-2,-1,0,1}

Range: {5,6}

Look at the first numbers or the

x-values in the ordered pairs

Since all the x-values are different, the set of ordered pairs is a function

List the domain or x-values

as {-2,-1,0,1} because we list the numbers in ascending order

List the range or y-values

as {5,6} because we list each number just once and in ascending order

We can also determine whether a graph represents a relation or a function by looking at the graph’s ordered pairs. In fact, we can tell whether a graph is a function or a relation by just looking at the graph. This is called the vertical line test.

If a vertical line passes through more than one point of a graph, then it is a **relation**.

If a vertical line does not pass through more than one point of a graph, then it is a **function**.

Reasoning: All ordered pairs on the same vertical line have the same x-value.

**Example: Identify each graph as a function or a relation. **

**Relation**

**Reasoning**

The graph is a relation because we can draw a vertical line through more than one point

**Function**

**Reasoning**

The graph is a function because any vertical line would pass through only one point

Functions can also be expressed as equations. Because of that, we can find values of functions. The equations that we will work with will be of the form

y = ax + b and y = ax2 + bx + c.

When we are asked to find the value of a function, we express the functions as f(x) = ax + b and f(x) = ax2 + bx + c, where f(x) = y and tells us to find the value of the function (y) for the given value of x.

f(x) is read "f of x".

**Examples: If f(x) = 2x - 5, find:**

- f(2)

f(2) = 2(2) - 5

f(2) = 4 - 5

f(2) = -1

f(2) tells us to substitute 2 in for x and then simplify

In 2(2) - 5, we multiply first to

get 4 - 5 then add to get -1 - f(-3)

f(-3) = 2(-3) - 5

f(-3) = -6 - 5

f(-3) = -11

f(-3) tells us to substitute -3 in for x and then simplify

In 2(-3) - 5, we multiply first to

get -6 - 5 and then add to get -11

By finding the values of the function, we can express a set of ordered pairs because f(x) = y.

Since f(2) = -1, y = -1 for (2,-1)

Since f(-3) = -11, y = -11 for (-3,-11)

**Remember:** When raising a number to a power, the exponent tells us the sign of our answer. If the exponent is even, then the answer is positive. If the exponent is odd, then the answer is negative.

**Example: If f(x) = x2 - 3x + 2, find **

- f(-1)

f(-1) = (-1)2 - 3(-1) + 2

f(-1) = 1 + 3 + 2

f(-1) = 6

f(-1) tells us to substitute -1 in for x

In (-1)2 - 3(-1) + 2, order of operations tells us to do the power and multiplication first 1 + 3 + 2 and then add to get 6 - f(4)

f(4) = (4)2 - 3(4) + 2

f(4) = 16 - 12 + 2

f(4) = 6

f(4) tells us to substitute 4 in for x and then simplify

In (4)2 - 3(4) + 2, order of operations tells us to do the power and multiplication first 16 - 12 + 2 and then add to get 6

A solution of a function is an ordered pair that makes a true sentence. As a result, we can tell whether an ordered pair is a solution by substituting the ordered pair into the equation and determining whether it makes a true sentence.

**Example: Determine whether the following ordered pairs is a solution to y = 3x - 5:**

- (-3,4)

4 = 3(-3) - 5

4 = -9 - 5

4 ≠ -14

(-3,4) is not a solution

Substitute the 4 in for y and

the -3 in for x

Simplify by multiplying first

3(-3) - 5 = -9 - 5 then adding

-9 - 5 = -14

4 is not equal to -14, so (-3,4) is not a solution - (4,7)

7=3(4)-5

7=12-5

7=7

(4,7) is a solution

substitute the 7 in for y and

the 4 in for x

Simplify by multiplying first

3(4) - 5 = 12 - 5 then adding

12 - 5 = 7

7 is equal to 7, so (4,7) is a solution

**Identify if each set of ordered pairs is a relation or a function. List the domain and range.**

- {(-3,8),(-2,5),(3,7),(5,-1)}
- {(4,3),(5,-2),(4,-4),(5,7)}
**Identify the following graphs as a relation or a function.**

**Identify if each set of ordered pairs is a relation or a function. List the domain and range.**

- {(-3,8),(-2,5),(3,7),(5,-1)}
- {(4,3),(5,-2),(4,-4),(5,7)}

Identify the following graphs as a relation or a function.

**Do the following tables represent a relation or a function? Express the table as a set of ordered pairs. List the domain and range.**-

Problem Solving - If you have pieces of paper in one box with the numbers 0, 1, 2, 3, 4 and pieces of paper in a second box with the numbers 5, 6, 7, 8, 9, explain how you could form five ordered pairs that would represent a function. Explain your reasoning.

- Function

Domain: {-3,-2,3,5}

Range: {-1,5,7,8} - Relation

Domain: {4,5}

Range: {-4,-2,3,7} - Function
- Relation
- Relation
- Function
- Relation

{(-5,7),(3,7),(-5,2),(3,2)}

Domain: {-5,3}

Range: {2,7} - To make sure that I would have five ordered pairs that represent a function, I would draw a number from the first box and then a number from the second box and pair them together. I would continue to do this until all five numbers from the first box are paired with a number from the second box.

- f(-2) =
- f(0) =
- f(-2) =
- f(0) =
- A = (-2,7)
- B = (3,6)

- f(-2) = 2(-2) - 5

f(-2) = -4 - 5

f(-2) = -9 - f(0) = 2(0) - 5

f(0) = 0 - 5

f(0) = -5 - f(-2) = 2(-2)2 - 4(-2) + 7

f(-2) = 2(4) + 8 + 7

f(-2) = 8 + 8 + 7

f(-2) = 23 - f(0) = 2(0)2 - 4(0) + 7

f(0) = 2(0) + 0 + 7

f(0) = 0 + 0 + 7

f(0) = 7 - A = (-2,7)

7 = -2(-2) + 3

7 = 4 + 3

7 = 7

(-2,7) is a solution - B = (3,6)

6 = -2(3) + 3

6 = -6 + 3

6 ≠ -3

(3,6) is not a solution - D = (1,2)

2 = (1)2 - 2(1) + 3

2 = 1 - 2 + 3

2 = 2

(1,2) is a solution - E = (-2,11)

11 = (-2)2 - 2(-2) + 3

11 = 4 + 4 + 3

11 = 11

(-2,11) is a solution

**Determine if each ordered pair is a solution for y = -2x + 3: **

**Determine if each ordered pair is a solution for y = x2 - 2x + 3:**

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