Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine whether a relation is a function, we need to check that each [tex]\( x \)[/tex]-value in the ordered pairs [tex]\( (x, y) \)[/tex] maps to exactly one [tex]\( y \)[/tex]-value. That is, no [tex]\( x \)[/tex]-value should be repeated with a different [tex]\( y \)[/tex].
Given the relation:
[tex]\[ R = \{(1,4), (1,3), (-1,3), (2,15)\} \][/tex]
Let's analyze each step carefully:
1. List the ordered pairs and identify [tex]\( x \)[/tex]-values:
[tex]\[ \begin{aligned} &\text{Pair } (1, 4) \to x = 1, \\ &\text{Pair } (1, 3) \to x = 1, \\ &\text{Pair } (-1, 3) \to x = -1, \\ &\text{Pair } (2, 15) \to x = 2. \end{aligned} \][/tex]
2. Check for repeated [tex]\( x \)[/tex]-values:
- The [tex]\( x \)[/tex]-value 1 appears in two different ordered pairs: [tex]\((1,4)\)[/tex] and [tex]\((1, 3)\)[/tex].
- Therefore, the pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] have the same [tex]\( x \)[/tex]-value, which violates the definition of a function. There are no other [tex]\( x \)[/tex]-values that repeat with different [tex]\( y \)[/tex]-values.
3. Double-check other ordered pairs:
- The pair [tex]\((-1, 3)\)[/tex] has an [tex]\( x \)[/tex]-value of -1, only appears once.
- The pair [tex]\((2, 15)\)[/tex] has an [tex]\( x \)[/tex]-value of 2, only appears once.
Next, let's address the pairs that have the same [tex]\( y \)[/tex]-value:
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] both map to the [tex]\( y \)[/tex]-value of 3. This is allowed and does not prevent the relation from being a function since there is no restriction that distinct [tex]\( x \)[/tex]-values must have distinct [tex]\( y \)[/tex]-values in a function. The critical criterion is that same [tex]\( x \)[/tex]-values should not map to different [tex]\( y \)[/tex]-values.
Thus, the ordered pairs that prevent this relation from being a function are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
Also, reviewing the problematic pair and ensuring correctness:
- The ordered pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] prevent the relation from being a function because they have the same [tex]\( x \)[/tex]-value.
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] having the same [tex]\( y \)[/tex]-value do not affect the definition of a function.
Conclusively, the correct pairs preventing the relation from being a function are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
Given the relation:
[tex]\[ R = \{(1,4), (1,3), (-1,3), (2,15)\} \][/tex]
Let's analyze each step carefully:
1. List the ordered pairs and identify [tex]\( x \)[/tex]-values:
[tex]\[ \begin{aligned} &\text{Pair } (1, 4) \to x = 1, \\ &\text{Pair } (1, 3) \to x = 1, \\ &\text{Pair } (-1, 3) \to x = -1, \\ &\text{Pair } (2, 15) \to x = 2. \end{aligned} \][/tex]
2. Check for repeated [tex]\( x \)[/tex]-values:
- The [tex]\( x \)[/tex]-value 1 appears in two different ordered pairs: [tex]\((1,4)\)[/tex] and [tex]\((1, 3)\)[/tex].
- Therefore, the pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] have the same [tex]\( x \)[/tex]-value, which violates the definition of a function. There are no other [tex]\( x \)[/tex]-values that repeat with different [tex]\( y \)[/tex]-values.
3. Double-check other ordered pairs:
- The pair [tex]\((-1, 3)\)[/tex] has an [tex]\( x \)[/tex]-value of -1, only appears once.
- The pair [tex]\((2, 15)\)[/tex] has an [tex]\( x \)[/tex]-value of 2, only appears once.
Next, let's address the pairs that have the same [tex]\( y \)[/tex]-value:
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] both map to the [tex]\( y \)[/tex]-value of 3. This is allowed and does not prevent the relation from being a function since there is no restriction that distinct [tex]\( x \)[/tex]-values must have distinct [tex]\( y \)[/tex]-values in a function. The critical criterion is that same [tex]\( x \)[/tex]-values should not map to different [tex]\( y \)[/tex]-values.
Thus, the ordered pairs that prevent this relation from being a function are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
Also, reviewing the problematic pair and ensuring correctness:
- The ordered pairs [tex]\((1, 4)\)[/tex] and [tex]\((1, 3)\)[/tex] prevent the relation from being a function because they have the same [tex]\( x \)[/tex]-value.
- The pairs [tex]\((1, 3)\)[/tex] and [tex]\((-1, 3)\)[/tex] having the same [tex]\( y \)[/tex]-value do not affect the definition of a function.
Conclusively, the correct pairs preventing the relation from being a function are:
[tex]\[ (1, 4) \quad \text{and} \quad (1, 3) \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
