Math Chapter 8

Question 1

Relation

Answer 1

A relation is a set of ordered pairs where the first coordinate represents the input (domain) and the second coordinate represents the output (range).

Question 2

Domain

Answer 2

The domain of a relation is the set of all first coordinates (inputs) in the ordered pairs.

Question 3

Range

Answer 3

The range of a relation is the set of all second coordinates (outputs) in the ordered pairs.

Question 4

Function

Answer 4

A function is a relation in which each element of the domain is paired with exactly one element of the range.

Question 5

Mapping Diagram

Answer 5

A mapping diagram is a visual representation that shows how elements of the domain are paired with elements of the range, helping to determine if a relation is a function.

Question 6

Is the relation {(-2,3), (2,2), (2,-2)} a function? Explain.

Answer 6

No, this is not a function because the input 2 is mapped to two different outputs (2 and -2). A function must have exactly one output for each input.

Question 7

Is the relation {(-5,-4), (0,-4), (5,-4)} a function? Explain.

Answer 7

Yes, this is a function because each input (-5, 0, 5) has exactly one output (-4). Even though all outputs are the same, each input has only one corresponding output.

Question 8

Is the relation given in Table 5 a function? Explain.

x | y |
|—-|—-|
| 5 | -3 |
| 4 | -3 |
| 1 | -2 |
| 0 | -2 |
| 7 | -5 |

Answer 8

Yes, this is a function. No x-values are repeated, and each input has exactly one output.

Question 9

Is the relation given in Table 6 a function? Explain.

x | y |
|—-|—-|
| 1 | 5 |
| 0 | 5 |
| 3 | 3 |
| 6 | -1 |
| 4 | 6 |

Answer 9

No, this is not a function. The x-value 0 appears twice with different y-values (5 and 5). A function cannot have multiple outputs for the same input.

Question 10

Is the relation given in Table 7 a function? Explain.

x | y |
|—-|—-|
| 2 | -4 |
| 1 | -3 |
| 0 | 0 |
| 4 | 4 |
| 4 | -5 |

Answer 10

No, this is not a function. The x-value 4 appears twice with different y-values (4 and -5). This means it fails the vertical-line test.

Question 11

Is age a function of height?

Answer 11

No, age is not a function of height. People of the same age can have different heights, meaning there are multiple outputs for a single input, which violates the definition of a function.

Question 12

Is the time you take to walk to the library a function of the distance to the library?

Answer 12

No, time is not necessarily a function of distance. Different individuals may take different amounts of time to walk the same distance based on speed, obstacles, or other factors. Since a single input (distance) can lead to multiple outputs (time taken), this is not a function.

Question 13

Is the price of a piece of cloth a function of the length of the cloth?

Answer 13

Yes, the price of cloth is typically a function of its length. Generally, the price is determined by its length (e.g., a fixed price per meter), meaning that for each length, there is exactly one price, satisfying the definition of a function.

Question 14

Find the solution of y = 3x + 4 for x = -1.

Answer 14

y = 3x + 4
y = 3(-1) + 4
y = -3 + 4
y = 1
A solution of the equation is (-1,1).

Question 15

The equation t = 21 - 0.01h models the normal low July temperature at Mt. Rushmore. Find the normal low July temperature at 300 m above the base.

Answer 15

t = 21 - 0.01(300)
t = 21 - 3
t = 18
A solution of the equation is (300, 18). The normal low July temperature at 300 m above the base is 18°C.

Question 16

Graph y = -2x + 3 and make a table of values.

Answer 16

Table of values:
| x | y |
|—-|—-|
| -2 | 7 |
| -1 | 5 |
| 0 | 3 |
| 1 | 1 |
| 2 | -1 |
Graph the ordered pairs and draw a line through the points.

Question 17

Graph each equation and determine if it is a function:
a) y = 2
b) x = 2

Answer 17

a) y = 2 → Horizontal line → It is a function.
b) x = 2 → Vertical line → It is not a function.

Question 18

Solve 3x + y = 5 for y and make a table of values.

Answer 18

Solve for y: y = 5 - 3x
Table of values:
| x | y |
|—-|—-|
| -2 | 1 |
| -1 | 2 |
| 0 | 5 |
| 1 | 2 |
| 2 | -1 |
Graph the equation.

Question 19

Find the solution of y = 3x + 4 for x = -1.

Answer 19

y = 3x + 4
y = 3(-1) + 4
y = -3 + 4
y = 1
A solution of the equation is (-1,1).

Question 20

The equation t = 21 - 0.01h models the normal low July temperature at Mt. Rushmore. Find the normal low July temperature at 300 m above the base.

Answer 20

t = 21 - 0.01(300)
t = 21 - 3
t = 18
A solution of the equation is (300, 18). The normal low July temperature at 300 m above the base is 18°C.

Question 21

Graph y = -2x + 3 and make a table of values.

Answer 21

Table of values:
| x | y |
|—-|—-|
| -2 | 7 |
| -1 | 5 |
| 0 | 3 |
| 1 | 1 |
| 2 | -1 |
Graph the ordered pairs and draw a line through the points.

Question 22

Graph each equation and determine if it is a function:
a) y = 2
b) x = 2

Answer 22

a) y = 2 → Horizontal line → It is a function.
b) x = 2 → Vertical line → It is not a function.

Question 23

Solve 3x + y = 5 for y and make a table of values.

Answer 23

Solve for y: y = 5 - 3x
Table of values:
| x | y |
|—-|—-|
| -2 | 1 |
| -1 | 2 |
| 0 | 5 |
| 1 | 2 |
| 2 | -1 |
Graph the equation.

Question 24

Question

Answer 24

Answer and Explanation

Question 25

Paulo works at a local store. Each week he earns a $300 salary plus a 3% commission on his sales.

a) Write a function rule that relates total earnings to sales.
b) Find his earnings for one week if his sales are $2,500.

Answer 25

a) Function Rule:
Let s = the amount of his sales.
Let t(s) = total earnings, a function of his sales.
t(s) = 300 + 0.03s

b) Finding earnings when sales = $2,500:
t(2,500) = 300 + 0.03(2,500)
t(2,500) = 300 + 75
t(2,500) = 375

Paulo earns $375 if his sales are $2,500.