2-1 Relations and Functions

Quick Review

A relation is a set of ordered pairs. The domain of a relation is the set of x-coordinates. The range is the set of y-coordinates. When each element of the domain is paired with exactly one element of the range, the relation is a function.

Example

Determine whether the relation is a function. Find the domain and range.

{(5, 0), (8, 1), (1, 3), (5, 2), (3, 8)}

In this relation, the x-coordinate 5 is paired with both 0 and 2. This relation is not a function.

The domain is the set of x-coordinates, which is {5, 8, 1, 3}.

The range is the set of y-coordinates, which is {0, 1, 3, 2, 8}.

Exercises

Determine whether each relation is a function. Find the domain and range.

3. {(10, 2), ( − 10 , 2 ) , negative 10 comma 2 close comma  (6, 4), (5, 3), ( − 6 , 7 ) negative 6 comma 7 close }
4. {(4, 5), (1, 5), (3, 8), (4, 6), (10, 12)}
5. 
6. 

For each function, find f ( − 2 ) , f ( − 0.5 ) , f open negative 2 close comma f open negative 0.5 close comma  and f(3).

7. f ( x ) = − x + 4 f open x close equals negative x plus 4
8. f ( x ) = 3 8 x − 3 f open x close equals , 3 eighths , x minus 3

2-2 Direct Variation

Quick Review

A linear equation of the form y = k x , k ≠ 0 , y equals k x comma k not equal to 0 comma  represents direct variation. The constant of variation is k. You can use proportions to solve direct variation problems.

Example

In the table, determine whether y varies directly with x. If so, what is the constant of variation and the function rule?

6 2 = 9 3 = 24 8 = 3 , 6 halves , equals , 9 thirds , equals , 24 over 8 , equals 3 . comma  so y varies directly with x, and the constant of variation is 3.

The function rule is y = 3 x . y equals 3 x .

Exercises

For each function, determine whether y varies directly with x. If so, find the constant of variation and write the function rule.

9. x	y
   − 2 negative 2	3
   1	4
   2	7

10. x	y
    4	5
    6	9
    10	17

11. x	y
    1	1
    2	2
    5	5

For each function, y varies directly with x. Find each constant of variation. Then find the value of y when x = − 0.3. x equals negative , 0.3.

12. y = 2 y equals 2  when x = − 1 2 x equals negative , 1 half
13. y = 2 3 y equals , 2 thirds  when x = 0.2 x equals 0.2
14. y = 7 y equals 7  when x = 2 x equals 2
15. y = 4 y equals 4  when x = − 3 x equals negative 3

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