The Function F ( X F(x F ( X ] Is Given By The Set Of Ordered Pairs: { ( 1 , 0 ) , ( − 10 , 2 ) , ( 0 , 6 ) , ( 3 , 17 ) , ( − 2 , − 1 ) } \{(1,0),(-10,2),(0,6),(3,17),(-2,-1)\} {( 1 , 0 ) , ( − 10 , 2 ) , ( 0 , 6 ) , ( 3 , 17 ) , ( − 2 , − 1 )} .Which Equation Is True?A. F ( − 10 ) = 1 F(-10)=1 F ( − 10 ) = 1 B. F ( 2 ) = − 10 F(2)=-10 F ( 2 ) = − 10 C. F ( 0 ) = 6 F(0)=6 F ( 0 ) = 6 D. F ( 1 ) = − 10 F(1)=-10 F ( 1 ) = − 10

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Introduction

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 often represented as a set of ordered pairs, where each pair consists of an input value and the corresponding output value. In this article, we will explore the function $f(x)$ given by the set of ordered pairs: $\{(1,0),(-10,2),(0,6),(3,17),(-2,-1)\}$. Our goal is to determine which equation is true among the given options.

Understanding Ordered Pairs

An ordered pair is a pair of values, usually written in the form $(x, y)$, where $x$ is the input value and $y$ is the output value. In the given set of ordered pairs, we have:

• $(1,0)$: This means that when the input value is $1$, the output value is $0$.
• $(-10,2)$: This means that when the input value is $-10$, the output value is $2$.
• $(0,6)$: This means that when the input value is $0$, the output value is $6$.
• $(3,17)$: This means that when the input value is $3$, the output value is $17$.
• $(-2,-1)$: This means that when the input value is $-2$, the output value is $-1$.

Analyzing the Options

Now that we have a good understanding of the ordered pairs, let's analyze the given options:

A. $f(-10)=1$ B. $f(2)=-10$ C. $f(0)=6$ D. $f(1)=-10$

We need to determine which of these equations is true based on the given ordered pairs.

Option A: $f(-10)=1$

Looking at the ordered pairs, we see that $(-10,2)$ is one of the pairs. This means that when the input value is $-10$, the output value is $2$, not $1$. Therefore, option A is incorrect.

Option B: $f(2)=-10$

We do not have any ordered pair with an input value of $2$. The closest pair is $(-10,2)$, but the input value is $-10$, not $2$. Therefore, option B is incorrect.

Option C: $f(0)=6$

Looking at the ordered pairs, we see that $(0,6)$ is one of the pairs. This means that when the input value is $0$, the output value is indeed $6$. Therefore, option C is correct.

Option D: $f(1)=-10$

We do have an ordered pair with an input value of $1$, which is $(1,0)$. This means that when the input value is $1$, the output value is $0$, not $-10$. Therefore, option D is incorrect.

Conclusion

In conclusion, the correct equation is $f(0)=6$. This means that when the input value is $0$, the output value is indeed $6$, as given by the ordered pair $(0,6)$.

Understanding Functions and Ordered Pairs

Functions and ordered pairs are fundamental concepts in mathematics. Understanding how to work with ordered pairs and how to analyze functions is crucial for success in mathematics and other fields. By following the steps outlined in this article, you can develop a deeper understanding of functions and ordered pairs and improve your problem-solving skills.

Common Mistakes to Avoid

When working with ordered pairs and functions, it's essential to avoid common mistakes. Some of these mistakes include:

• Confusing input and output values
• Not paying attention to the order of the ordered pairs
• Not analyzing the given options carefully
• Not using the correct notation for ordered pairs

By being aware of these common mistakes, you can avoid them and improve your understanding of functions and ordered pairs.

Real-World Applications

Functions and ordered pairs have numerous real-world applications. Some examples include:

• Modeling population growth
• Analyzing data
• Creating algorithms
• Solving optimization problems

By understanding functions and ordered pairs, you can develop a deeper understanding of these real-world applications and improve your problem-solving skills.

Final Thoughts

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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 often represented as a set of ordered pairs, where each pair consists of an input value and the corresponding output value.

Q: What is an ordered pair?

A: An ordered pair is a pair of values, usually written in the form $(x, y)$, where $x$ is the input value and $y$ is the output value.

Q: How do I determine if an equation is true for a function?

A: To determine if an equation is true for a function, you need to check if the input value is in the domain of the function and if the output value matches the given value. You can do this by looking at the ordered pairs of the function.

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

A: A function is a relation where each input value corresponds to exactly one output value. A relation, on the other hand, can have multiple output values for the same input value.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to look at the input values of the ordered pairs. The domain is the set of all possible input values.

Q: How do I find the range of a function?

A: To find the range of a function, you need to look at the output values of the ordered pairs. The range is the set of all possible output values.

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

A: A function is a relation between a set of inputs and a set of outputs, while an equation is a statement that two expressions are equal. A function can be represented as an equation, but not all equations are functions.

Q: How do I determine if a function is one-to-one or many-to-one?

A: To determine if a function is one-to-one or many-to-one, you need to check if each input value corresponds to exactly one output value. If each input value corresponds to exactly one output value, the function is one-to-one. If multiple input values correspond to the same output value, the function is many-to-one.

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

A: A function is a relation between a set of inputs and a set of outputs, while a graph is a visual representation of the function. A graph can be used to visualize the function, but it is not the function itself.

Q: How do I use functions in real-world applications?

A: Functions are used in many real-world applications, such as modeling population growth, analyzing data, creating algorithms, and solving optimization problems. By understanding functions, you can develop a deeper understanding of these applications and improve your problem-solving skills.

Q: What are some common mistakes to avoid when working with functions?

A: Some common mistakes to avoid when working with functions include:

• Confusing input and output values
• Not paying attention to the order of the ordered pairs
• Not analyzing the given options carefully
• Not using the correct notation for ordered pairs

By being aware of these common mistakes, you can avoid them and improve your understanding of functions.

Q: How can I practice working with functions?

A: You can practice working with functions by:

• Creating your own functions and ordered pairs
• Analyzing functions and determining if they are one-to-one or many-to-one
• Using functions to solve real-world problems
• Working with different types of functions, such as linear and quadratic functions

By practicing working with functions, you can develop a deeper understanding of this concept and improve your problem-solving skills.