Find The Range Of The Function Defined By The Table Below. Express Your Answer As A Set Of Numbers.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 5 & -7 \\ \hline -10 & 6 \\ \hline 1 & -7 \\ \hline \end{tabular} \\]

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Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. When a function is defined by a table, finding the range can be a bit tricky, but it's still a crucial concept to understand. In this article, we'll explore how to find the range of a function defined by a table and provide a step-by-step guide on how to do it.

Understanding the Table

Before we dive into finding the range, let's take a closer look at the table provided.

xx yy
5 -7
-10 6
1 -7

From the table, we can see that there are three input-output pairs: (5, -7), (-10, 6), and (1, -7). These pairs represent the possible input values and their corresponding output values.

Finding the Range

To find the range, we need to identify all the unique output values in the table. In this case, we have two unique output values: -7 and 6.

However, we need to consider the possibility that there may be other output values that are not listed in the table. To ensure that we have found all the possible output values, we need to analyze the function's behavior.

Analyzing the Function's Behavior

Let's examine the table again and look for any patterns or relationships between the input and output values.

xx yy
5 -7
-10 6
1 -7

From the table, we can see that the output value -7 is associated with two different input values: 5 and 1. This suggests that the function may have a constant output value for certain input values.

On the other hand, the output value 6 is associated with only one input value: -10. This suggests that the function may have a unique output value for this input value.

Conclusion

Based on our analysis, we can conclude that the range of the function defined by the table is the set of all possible output values. In this case, the range is {-7, 6}.

However, we need to consider the possibility that there may be other output values that are not listed in the table. To ensure that we have found all the possible output values, we need to analyze the function's behavior.

Step-by-Step Guide to Finding the Range

Here's a step-by-step guide to finding the range of a function defined by a table:

  1. Identify the input-output pairs: Look at the table and identify the input-output pairs.
  2. Find the unique output values: Identify the unique output values in the table.
  3. Analyze the function's behavior: Examine the table and look for any patterns or relationships between the input and output values.
  4. Consider the possibility of other output values: Consider the possibility that there may be other output values that are not listed in the table.
  5. Determine the range: Based on your analysis, determine the range of the function.

Example 2

Let's consider another example to illustrate the concept of finding the range of a function defined by a table.

Suppose we have the following table:

xx yy
2 3
4 5
6 7

To find the range, we need to follow the same steps as before.

  1. Identify the input-output pairs: Look at the table and identify the input-output pairs.
  2. Find the unique output values: Identify the unique output values in the table.
  3. Analyze the function's behavior: Examine the table and look for any patterns or relationships between the input and output values.
  4. Consider the possibility of other output values: Consider the possibility that there may be other output values that are not listed in the table.
  5. Determine the range: Based on your analysis, determine the range of the function.

In this case, the unique output values are 3, 5, and 7. However, we need to consider the possibility that there may be other output values that are not listed in the table.

After analyzing the function's behavior, we can conclude that the range of the function defined by the table is the set of all possible output values. In this case, the range is {3, 5, 7}.

Conclusion

In conclusion, finding the range of a function defined by a table requires careful analysis of the input-output pairs and the function's behavior. By following the step-by-step guide outlined in this article, you can determine the range of a function defined by a table.

Final Thoughts

Finding the range of a function defined by a table is an essential concept in mathematics. By understanding how to find the range, you can better analyze and understand the behavior of functions. Whether you're a student or a professional, having a solid grasp of this concept can help you tackle complex mathematical problems with confidence.

References

Q: What is the range of a function?

A: The range of a function is the set of all possible output values it can produce for the given input values.

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

A: To find the range of a function defined by a table, follow these steps:

  1. Identify the input-output pairs: Look at the table and identify the input-output pairs.
  2. Find the unique output values: Identify the unique output values in the table.
  3. Analyze the function's behavior: Examine the table and look for any patterns or relationships between the input and output values.
  4. Consider the possibility of other output values: Consider the possibility that there may be other output values that are not listed in the table.
  5. Determine the range: Based on your analysis, determine the range of the function.

Q: What if there are multiple output values for the same input value?

A: If there are multiple output values for the same input value, it means that the function is not one-to-one, and the range will include all the unique output values.

Q: Can the range of a function be empty?

A: Yes, the range of a function can be empty. This occurs when there are no output values for any input values.

Q: How do I determine if the range of a function is finite or infinite?

A: To determine if the range of a function is finite or infinite, examine the table and look for any patterns or relationships between the input and output values. If the output values are bounded, the range is finite. If the output values are unbounded, the range is infinite.

Q: Can the range of a function be a single value?

A: Yes, the range of a function can be a single value. This occurs when the function always produces the same output value for any input value.

Q: How do I find the range of a function defined by an equation?

A: To find the range of a function defined by an equation, follow these steps:

  1. Solve the equation: Solve the equation to find the output values.
  2. Identify the unique output values: Identify the unique output values.
  3. Analyze the function's behavior: Examine the equation and look for any patterns or relationships between the input and output values.
  4. Consider the possibility of other output values: Consider the possibility that there may be other output values that are not listed in the equation.
  5. Determine the range: Based on your analysis, determine the range of the function.

Q: Can the range of a function be a set of numbers?

A: Yes, the range of a function can be a set of numbers. This occurs when the function produces a set of output values for any input value.

Q: How do I determine if the range of a function is an interval?

A: To determine if the range of a function is an interval, examine the table and look for any patterns or relationships between the input and output values. If the output values are bounded and include all the values between the minimum and maximum output values, the range is an interval.

Q: Can the range of a function be a union of intervals?

A: Yes, the range of a function can be a union of intervals. This occurs when the function produces a set of output values that are not continuous.

Conclusion

In conclusion, finding the range of a function defined by a table or an equation requires careful analysis of the input-output pairs and the function's behavior. By following the step-by-step guide outlined in this article, you can determine the range of a function and understand its behavior.

Final Thoughts

Finding the range of a function is an essential concept in mathematics. By understanding how to find the range, you can better analyze and understand the behavior of functions. Whether you're a student or a professional, having a solid grasp of this concept can help you tackle complex mathematical problems with confidence.

References