C. Does The Following Relation On $x$ And $y$ Make For A Function Of $x$?$\{(10,3),(-10,4),(-5,3),(-6,1)\}$- Yes, This Relation Describes A Function Of $x$.- No, This Relation Does Not Describe A
In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. For a relation to be considered a function, each input must correspond to exactly one output. In other words, for every value of x, there must be only one value of y.
Understanding Functions
A function is often represented as a set of ordered pairs, where each pair consists of an input value (x) and an output value (y). For example, the relation {(2, 3), (4, 5), (6, 7)} represents a function, as each input value (2, 4, 6) corresponds to exactly one output value (3, 5, 7).
Analyzing the Given Relation
The given relation is {(10, 3), (-10, 4), (-5, 3), (-6, 1)}. To determine whether this relation describes a function of x, we need to examine each ordered pair and check if each input value corresponds to exactly one output value.
Examining the Ordered Pairs
Let's examine each ordered pair in the given relation:
- (10, 3): In this pair, the input value x is 10, and the output value y is 3. This pair satisfies the condition of a function, as there is only one output value for the input value 10.
- (-10, 4): In this pair, the input value x is -10, and the output value y is 4. This pair also satisfies the condition of a function, as there is only one output value for the input value -10.
- (-5, 3): In this pair, the input value x is -5, and the output value y is 3. This pair satisfies the condition of a function, as there is only one output value for the input value -5.
- (-6, 1): In this pair, the input value x is -6, and the output value y is 1. This pair also satisfies the condition of a function, as there is only one output value for the input value -6.
Conclusion
Based on the analysis of each ordered pair, we can conclude that the given relation {(10, 3), (-10, 4), (-5, 3), (-6, 1)} does describe a function of x. Each input value corresponds to exactly one output value, satisfying the condition of a function.
Why is this Relation a Function?
This relation is a function because each input value (10, -10, -5, -6) corresponds to exactly one output value (3, 4, 3, 1). There are no duplicate input values, and each input value has a unique output value. This satisfies the definition of a function, which requires each input value to correspond to exactly one output value.
What if there were Duplicate Input Values?
If there were duplicate input values, the relation would not be a function. For example, if the relation were {(10, 3), (10, 4), (-5, 3), (-6, 1)}, it would not be a function because the input value 10 corresponds to two different output values (3 and 4).
Conclusion
In conclusion, the given relation {(10, 3), (-10, 4), (-5, 3), (-6, 1)} does describe a function of x. Each input value corresponds to exactly one output value, satisfying the condition of a function. This relation is a good example of a function, and it can be used to illustrate the concept of a function in mathematics.
Key Takeaways
- A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
- For a relation to be considered a function, each input value must correspond to exactly one output value.
- The given relation {(10, 3), (-10, 4), (-5, 3), (-6, 1)} does describe a function of x.
- Each input value corresponds to exactly one output value, satisfying the condition of a function.
Final Thoughts
In the previous article, we discussed whether the given relation {(10, 3), (-10, 4), (-5, 3), (-6, 1)} describes a function of x. We concluded that it does, as each input value corresponds to exactly one output value. In this article, we will answer some frequently asked questions about functions and relations.
Q: What is a function?
A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. For a relation to be considered a function, each input value must correspond to exactly one output value.
Q: What is the difference between a function and a relation?
A relation is a set of ordered pairs, where each pair consists of an input value and an output value. A function is a relation where each input value corresponds to exactly one output value. In other words, a function is a relation that satisfies the condition of a function.
Q: How do I determine if a relation is a function?
To determine if a relation is a function, you need to examine each ordered pair and check if each input value corresponds to exactly one output value. If there are any duplicate input values, the relation is not a function.
Q: What if there are duplicate input values?
If there are duplicate input values, the relation is not a function. For example, if the relation were {(10, 3), (10, 4), (-5, 3), (-6, 1)}, it would not be a function because the input value 10 corresponds to two different output values (3 and 4).
Q: Can a function have multiple output values?
No, a function cannot have multiple output values for a single input value. By definition, a function is a relation where each input value corresponds to exactly one output value.
Q: Can a relation have multiple input values?
Yes, a relation can have multiple input values. However, for a relation to be considered a function, each input value must correspond to exactly one output value.
Q: What is the domain of a function?
The domain of a function is the set of all input values. In other words, it is the set of all x-values.
Q: What is the range of a function?
The range of a function is the set of all output values. In other words, it is the set of all y-values.
Q: Can the domain and range of a function be the same?
Yes, the domain and range of a function can be the same. For example, if the function is f(x) = x, the domain and range are both the set of all real numbers.
Q: Can the domain and range of a function be different?
Yes, the domain and range of a function can be different. For example, if the function is f(x) = 1/x, the domain is the set of all non-zero real numbers, and the range is the set of all non-zero real numbers.
Q: What is the difference between a one-to-one function and a many-to-one function?
A one-to-one function is a function where each output value corresponds to exactly one input value. A many-to-one function is a function where multiple input values correspond to the same output value.
Q: Can a function be both one-to-one and many-to-one?
No, a function cannot be both one-to-one and many-to-one. By definition, a one-to-one function is a function where each output value corresponds to exactly one input value, while a many-to-one function is a function where multiple input values correspond to the same output value.
Conclusion
In conclusion, functions and relations are important concepts in mathematics. Understanding functions and relations is crucial for solving problems in mathematics, science, and engineering. In this article, we answered some frequently asked questions about functions and relations, and we hope that this information will be helpful to you.