Which Set Of Ordered Pairs Represents A Function?A. \[$\{(2,-2),(1,5),(-2,2),(1,-3),(8,-1)\}\$\]B. \[$\{(3,-1),(7,1),(-6,-1),(9,1),(2,-1)\}\$\]C. \[$\{(6,8),(5,2),(-2,-5),(1,-3),(-2,9)\}\$\]D.

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Understanding Functions in Mathematics

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between variables, where each input corresponds to exactly one output. In other words, for every input, there is only one output. This is a fundamental concept in mathematics, and it has numerous applications in various fields, including science, engineering, and economics.

Ordered Pairs and Functions

Ordered pairs are a way of representing relationships between two variables. They consist of an ordered pair of values, where the first value is the input and the second value is the output. For example, the ordered pair (2, 4) represents a relationship where the input is 2 and the output is 4. In the context of functions, ordered pairs are used to represent the input-output relationships.

Determining if a Set of Ordered Pairs Represents a Function

To determine if a set of ordered pairs represents a function, we need to check if each input corresponds to exactly one output. In other words, we need to check if there are any repeated inputs with different outputs. If there are, then the set of ordered pairs does not represent a function.

Analyzing the Options

Let's analyze the options given:

A. {(2,−2),(1,5),(−2,2),(1,−3),(8,−1)}\{(2,-2),(1,5),(-2,2),(1,-3),(8,-1)\}

This set of ordered pairs contains the following inputs: 2, 1, -2, 1, and 8. However, the input 1 is repeated with different outputs: 5 and -3. This means that the set of ordered pairs does not represent a function.

B. {(3,−1),(7,1),(−6,−1),(9,1),(2,−1)}\{(3,-1),(7,1),(-6,-1),(9,1),(2,-1)\}

This set of ordered pairs contains the following inputs: 3, 7, -6, 9, and 2. However, the input 2 is repeated with the same output: -1. This means that the set of ordered pairs represents a function.

C. {(6,8),(5,2),(−2,−5),(1,−3),(−2,9)}\{(6,8),(5,2),(-2,-5),(1,-3),(-2,9)\}

This set of ordered pairs contains the following inputs: 6, 5, -2, 1, and -2. However, the input -2 is repeated with different outputs: -5 and 9. This means that the set of ordered pairs does not represent a function.

D.

There is no option D.

Conclusion

In conclusion, the set of ordered pairs that represents a function is option B: {(3,−1),(7,1),(−6,−1),(9,1),(2,−1)}\{(3,-1),(7,1),(-6,-1),(9,1),(2,-1)\}. This set of ordered pairs contains no repeated inputs with different outputs, which means that each input corresponds to exactly one output. Therefore, it represents a function.

Key Takeaways

  • A function is a relation between a set of inputs and a set of possible outputs.
  • Ordered pairs are used to represent the input-output relationships.
  • To determine if a set of ordered pairs represents a function, we need to check if each input corresponds to exactly one output.
  • If there are any repeated inputs with different outputs, then the set of ordered pairs does not represent a function.

Real-World Applications

Functions have numerous applications in various fields, including science, engineering, and economics. For example, in physics, the position of an object as a function of time is a function that describes the motion of the object. In economics, the demand for a product as a function of its price is a function that describes the relationship between the price and the quantity demanded.

Practice Problems

  1. Determine if the following set of ordered pairs represents a function: {(4,2),(2,5),(1,3),(4,2),(6,1)}\{(4,2),(2,5),(1,3),(4,2),(6,1)\}.
  2. Determine if the following set of ordered pairs represents a function: {(8,3),(5,2),(−2,1),(8,3),(9,4)}\{(8,3),(5,2),(-2,1),(8,3),(9,4)\}.

Answer Key

  1. No, the set of ordered pairs does not represent a function.
  2. No, the set of ordered pairs does not represent a function.

References

  • "Functions" by Khan Academy
  • "Functions" by Math Open Reference
  • "Functions" by Wolfram MathWorld
    Q&A: Functions in Mathematics =====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about functions in mathematics.

Q: What is a function?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a way of describing a relationship between variables, where each input corresponds to exactly one output.

Q: What is the difference between a function and a relation?

A: A relation is a set of ordered pairs that describes a relationship between variables. A function is a special type of relation where each input corresponds to exactly one output.

Q: How do I determine if a set of ordered pairs represents a function?

A: To determine if a set of ordered pairs represents a function, you need to check if each input corresponds to exactly one output. If there are any repeated inputs with different outputs, then the set of ordered pairs does not represent a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible inputs. It is the set of all values that can be plugged into the function.

Q: What is the range of a function?

A: The range of a function is the set of all possible outputs. It is the set of all values that can be produced by the function.

Q: Can a function have multiple outputs for the same input?

A: No, a function cannot have multiple outputs for the same input. By definition, a function is a relation where each input corresponds to exactly one output.

Q: Can a function have no outputs for a given input?

A: Yes, a function can have no outputs for a given input. This is known as a "hole" in the function.

Q: What is the difference between a function and a graph?

A: A function is a mathematical relation between inputs and outputs, while a graph is a visual representation of the function.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on a coordinate plane. You can use a table of values or a graphing calculator to help you.

Q: What are some common types of functions?

A: Some common types of functions include:

  • Linear functions
  • Quadratic functions
  • Polynomial functions
  • Rational functions
  • Trigonometric functions

Q: How do I evaluate a function?

A: To evaluate a function, you need to plug in the input value and perform the necessary operations.

Q: What is the difference between a function and an equation?

A: A function is a relation between inputs and outputs, while an equation is a statement that two expressions are equal.

Q: Can a function be an equation?

A: Yes, a function can be an equation. In fact, many functions can be represented as equations.

Q: What is the importance of functions in mathematics?

A: Functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics.

Q: How do I use functions in real-life situations?

A: Functions are used in many real-life situations, such as:

  • Modeling population growth
  • Describing the motion of objects
  • Analyzing data
  • Making predictions

Conclusion

In conclusion, functions are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding functions, you can better analyze and solve problems in mathematics and real-life situations.

Key Takeaways

  • A function is a relation between inputs and outputs.
  • A function can be represented as an equation.
  • Functions have numerous applications in various fields.
  • Functions can be used to model real-life situations.

Practice Problems

  1. Determine if the following set of ordered pairs represents a function: {(4,2),(2,5),(1,3),(4,2),(6,1)}\{(4,2),(2,5),(1,3),(4,2),(6,1)\}.
  2. Evaluate the function f(x)=2x+3f(x) = 2x + 3 at x=4x = 4.
  3. Graph the function f(x)=x2+1f(x) = x^2 + 1.

Answer Key

  1. No, the set of ordered pairs does not represent a function.
  2. f(4)=2(4)+3=11f(4) = 2(4) + 3 = 11.
  3. The graph of the function is a parabola that opens upward.

References

  • "Functions" by Khan Academy
  • "Functions" by Math Open Reference
  • "Functions" by Wolfram MathWorld