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Which of the relations given by the following sets of ordered pairs is a function?

A. [tex]$\{(1,2),(2,3),(3,4),(2,1),(1,0)\}$[/tex]

B. [tex]$\{(2,-8),(6,4),(-3,9),(2,0),(-5,3)\}$[/tex]

C. [tex]$\{(1,-3),(1,-1),(1,1),(1,3),(1,5)\}$[/tex]

D. [tex]$\{(-2,5),(7,5),(-4,0),(3,1),(0,-6)\}$[/tex]

Sagot :

GinnyAnswer
To determine which of the given sets of ordered pairs represents a function, we need to understand what defines a function. A relation (set of ordered pairs) is a function if each input (first element of the ordered pair) has exactly one output (second element of the ordered pair). In other words, no two pairs should have the same first element if they are to be considered a function.

Let's analyze each set:

1. Set 1: [tex]$\{(1,2),(2,3),(3,4),(2,1),(1,0)\}$[/tex]
- Here, we have the ordered pairs: [tex]$(1,2)$[/tex], [tex]$(2,3)$[/tex], [tex]$(3,4)$[/tex], [tex]$(2,1)$[/tex], and [tex]$(1,0)$[/tex].
- Notice that the input value [tex]$1$[/tex] is paired with both [tex]$2$[/tex] and [tex]$0$[/tex], and the input value [tex]$2$[/tex] is paired with both [tex]$3$[/tex] and [tex]$1$[/tex].
- Since there are input values that have multiple outputs, this set is not a function.

2. Set 2: [tex]$\{(2,-8),(6,4),(-3,9),(2,0),(-5,3)\}$[/tex]
- Here, we have the ordered pairs: [tex]$(2,-8)$[/tex], [tex]$(6,4)$[/tex], [tex]$(-3,9)$[/tex], [tex]$(2,0)$[/tex], and [tex]$(-5,3)$[/tex].
- Notice that the input value [tex]$2$[/tex] is paired with both [tex]$-8$[/tex] and [tex]$0$[/tex].
- Since there is an input value that has multiple outputs, this set is not a function.

3. Set 3: [tex]$\{(1,-3),(1,-1),(1,1),(1,3),(1,5)\}$[/tex]
- Here, we have the ordered pairs: [tex]$(1,-3)$[/tex], [tex]$(1,-1)$[/tex], [tex]$(1,1)$[/tex], [tex]$(1,3)$[/tex], and [tex]$(1,5)$[/tex].
- Notice that the input value [tex]$1$[/tex] is paired with multiple outputs: [tex]$-3$[/tex], [tex]$-1$[/tex], [tex]$1$[/tex], [tex]$3$[/tex], and [tex]$5$[/tex].
- Since there is an input value that has multiple outputs, this set is not a function.

4. Set 4: [tex]$\{(-2,5),(7,5),(-4,0),(3,1),(0,-6)\}$[/tex]
- Here, we have the ordered pairs: [tex]$(-2,5)$[/tex], [tex]$(7,5)$[/tex], [tex]$(-4,0)$[/tex], [tex]$(3,1)$[/tex], and [tex]$(0,-6)$[/tex].
- Each input value is paired with exactly one output: [tex]$-2 \rightarrow 5$[/tex], [tex]$7 \rightarrow 5$[/tex], [tex]$-4 \rightarrow 0$[/tex], [tex]$3 \rightarrow 1$[/tex], and [tex]$0 \rightarrow -6$[/tex].
- Since each input value has exactly one output, this set is a function.

In summary, only relation 4 (the fourth set of ordered pairs) represents a function.
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