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To further understand functions, let's study the following.

Given the following ordered pairs, which relations are functions?

[tex]\[
\begin{array}{l}
A=\{(1,2),(2,3),(3,4),(4,5)\} \\
B=\{(3,3),(4,4),(5,5),(6,6)\} \\
C=\{(1,0),(0,1),(-1,0),(0,-1)\} \\
D=\{(a,b),(b,c),(c,d),(a,d)\}
\end{array}
\][/tex]


Sagot :

To determine which of the given relations are functions, we need to verify if each relation maps every input to exactly one unique output. In simpler terms, for a set of ordered pairs to represent a function, each first element (input) must be associated with exactly one second element (output).

Let's go through each relation step-by-step:

### Relation [tex]\( A = \{(1, 2), (2, 3), (3, 4), (4, 5)\} \)[/tex]
- Inputs: 1, 2, 3, 4
- Outputs: 2, 3, 4, 5

In relation A, each input value is unique and maps to exactly one output value. Therefore, [tex]\( A \)[/tex] is a function.

### Relation [tex]\( B = \{(3, 3), (4, 4), (5, 5), (6, 6)\} \)[/tex]
- Inputs: 3, 4, 5, 6
- Outputs: 3, 4, 5, 6

In relation B, each input value is unique and maps to exactly one output value. Therefore, [tex]\( B \)[/tex] is a function.

### Relation [tex]\( C = \{(1, 0), (0, 1), (-1, 0), (0, -1)\} \)[/tex]
- Inputs: 1, 0, -1, 0
- Outputs: 0, 1, 0, -1

In relation C, the input value 0 appears twice (in the pairs [tex]\((0, 1)\)[/tex] and [tex]\((0, -1)\)[/tex]) and maps to different outputs (1 and -1). Therefore, [tex]\( C \)[/tex] is not a function.

### Relation [tex]\( D = \{(a, b), (b, c), (c, d), (a, d)\} \)[/tex]
- Inputs: a, b, c, a
- Outputs: b, c, d, d

In relation D, the input value [tex]\(a\)[/tex] appears twice (in the pairs [tex]\((a, b)\)[/tex] and [tex]\((a, d)\)[/tex]) and maps to different outputs (b and d). Therefore, [tex]\( D \)[/tex] is not a function.

### Conclusion
From our analysis, the relations that are functions are:
- Relation [tex]\( A \)[/tex]
- Relation [tex]\( B \)[/tex]

Thus, the functions are [tex]\( A \)[/tex] and [tex]\( B \)[/tex].