### Key Concepts

• Identifying the functions from the arrow diagram

• Usage of tables to identifying functions

• Interpretation of Functions

## Introduction:

- In this chapter, we will learn to identify the functions from the arrow diagrams.

- Use tables to identify functions and interpretation of functions in the given scenario.

In the earlier chapter, we learned about functions and relations.

### What is a Set?

A set is a collection of distinct or well-defined members or elements.

For example,

{a, b, c, …, x, y, z} is a set of alphabet letters.

### What are Ordered-pair Numbers?

These are numbers that go hand in hand.

For example, (7, 9) is an ordered-pair number whereby the numbers 7 and 9 are the first and second elements, respectively.

### What is the Relation?

A relation is a subset of the Cartesian product. In other words, we can define a relation as a bunch of ordered pairs.

Real-life relations:

(Brother, sister)

(Father, son)

(Teacher, student)

### What is a Function?

A function is a relation that describes that there should be only one output for each input.

### 3.1 Understand Relations and Functions:

In many naturally occurring phenomena, two variables may be linked by some type of relationship. For instance, an archeologist finds the bones of a woman at an excavation site. One of the bones is a femur. The femur is the large bone in the thigh attached to the knee and hip.

The following table shows a correspondence between the length of a woman’s femur and her height.

Length of Femur(cm) X | Height(in.) y | Ordered Pair |

45.5 | 65.5 | → (45.5, 65.5) |

48.2 | 68.0 | → (48.2, 68.0) |

41.8 | 62.2 | → (41.8, 62.2) |

46.0 | 66.0 | → (46.0, 66.0) |

50.4 | 70.0 | → (50.4, 70.0) |

For example, the following table depicts five states in the United States and the corresponding number of representatives in the House of Representatives as of July 2005.

State X | Number of Representatives Y |

Alabama | 7 |

California | 53 |

Colorado | 7 |

Florida | 25 |

Kansas | 4 |

These data define a relation:

{(Alabama, 7), (California, 53), (Colorado, 7), (Florida, 25), (Kansas, 4)}

### 3.1.1 Identify Functions with Arrow Diagrams

Jeo needs to advertise his company. He considers several different brochures of different side lengths and areas.

**Step 1:** Organize the given data using ordered pairs.

Input | output |

Side Length | Area |

4 | 24 |

5 | 35 |

6 | 42 |

7 | 49 |

9 | 72 |

**Step 2:** Match each input value to its output value by using an arrow diagram

In this example, each input is assigned exactly one output. So, the relation is a function.

### 3.1.2 Use Tables to Identify Functions

Jeo uses a table to record the sales of his products per week.

Week (x) | Sales (y) |

1 | 100 |

2 | 116 |

3 | 125 |

4 | 94 |

5 | 116 |

To determine whether this relation is a function or not, we need to find whether each input has exactly one output or not.

If we observe the values, week 2 and week 5 have the same number of sales.

Hence, this relation is not a function.

Let us see another example,

Look at the below table

X | y |

-3 | 7 |

-1 | 5 |

0 | -2 |

5 | 9 |

5 | 3 |

Since we have repetitions or duplicates of x-values with different y-values, then this relation ceases to be a function.

Let us see another example,

Look at the below table

X | Y |

-2 | 0 |

-1 | -2 |

0 | 3 |

4 | -1 |

5 | -3 |

This relation is definitely a function because every x-value is unique and is associated with only one value of *y*.

### 3.1.3 Interpretation of Functions

Jeo goes for a vacation trip and boards in a hotel

The room charges are displayed as follows:

Days | Charge ($) |

Upto 1 day | $28 |

Upto 2 days | $54 |

Upto 3 days | $80 |

Upto 4 days | $100 |

Upto 5 days | $120 |

- Is the cost to stay a function? Explain.

The charges are different for the number of days.

So, the cost to stay is a function.

- If Jeo wants to stay for more than 5 days, should Jeo expect to pay more than $120?

Yes, Jeo needs to pay more than $120.

## Exercise:

1. Which of the following arrow diagrams (a) and (b) represent functions? If one does not represent a function, explain why not.

2. Which of the following are functions from *x* to *y*? Assuming that the entire set of ordered pairs is given.

(a) {(1, 6), (2, 6), (3, 4), (4, 5)}

(b) {((1, 4),(5, 1),(5, 2),(7, 9)}

3. Check if the following ordered pairs are functions:

- W= {(1, 2), (2, 3), (3, 4), (4, 5)
- Y = {(1, 6), (2, 5), (1, 9), (4, 3)}

4. Look at the table below and identify whether it is a function or not? Explain.

5. Is the mapping diagram a relation or function? Explain

6. State whether each of the following relations represents a function or not.

7. State whether each of the following relations represents a function or not.

8. The table gives a relation between a person’s age and the person’s maximum recommended heart rate

*a. What is the domain?*

*b. What is the range?*

*c. The range element 200 corresponds to what element in the domain?*

*d. Complete the ordered pair: (50, )*

*e. Complete the ordered pair: ( , 190)*

9. Write the relation as a set of ordered pairs

10. Write the relation as a set of ordered pairs

### What have we learned:

• Relations

• Functions

• Identifying functions with arrow diagrams

• Using tables to identify functions

• Interpreting functions in the given scenario.

#### Concept Map:

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