Which Is The Function Represented By The Table? A 2-column Table With 4 Rows. Column 1 Is Labeled X With Entries Negative 1, 0, 1, 2. Column 2 Is Labeled F (x) With Entries 4, 5, 6, 7. F (x) = X 5 F (x) = X Minus 5 F (x) = X 1 F (x) = X Minus 1

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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 using a table, where the input values are listed in one column and the corresponding output values are listed in another column. In this article, we will explore the function represented by a 2-column table with 4 rows, and examine the different possibilities for the function.

The Table

x f(x)
-1 4
0 5
1 6
2 7

Possible Functions

There are four possible functions that can be represented by this table, each corresponding to a different equation:

f(x) = x^5

This function is a simple power function, where the output value is equal to the input value raised to the power of 5. To determine if this is the correct function, we can plug in the values from the table into the equation and see if the results match the output values.

  • For x = -1, f(x) = (-1)^5 = -1, which does not match the output value of 4.
  • For x = 0, f(x) = (0)^5 = 0, which does not match the output value of 5.
  • For x = 1, f(x) = (1)^5 = 1, which does not match the output value of 6.
  • For x = 2, f(x) = (2)^5 = 32, which does not match the output value of 7.

Therefore, this is not the correct function.

f(x) = x - 5

This function is a linear function, where the output value is equal to the input value minus 5. To determine if this is the correct function, we can plug in the values from the table into the equation and see if the results match the output values.

  • For x = -1, f(x) = (-1) - 5 = -6, which does not match the output value of 4.
  • For x = 0, f(x) = (0) - 5 = -5, which does not match the output value of 5.
  • For x = 1, f(x) = (1) - 5 = -4, which does not match the output value of 6.
  • For x = 2, f(x) = (2) - 5 = -3, which does not match the output value of 7.

Therefore, this is not the correct function.

f(x) = x^1

This function is a simple identity function, where the output value is equal to the input value. To determine if this is the correct function, we can plug in the values from the table into the equation and see if the results match the output values.

  • For x = -1, f(x) = (-1)^1 = -1, which does not match the output value of 4.
  • For x = 0, f(x) = (0)^1 = 0, which does not match the output value of 5.
  • For x = 1, f(x) = (1)^1 = 1, which does not match the output value of 6.
  • For x = 2, f(x) = (2)^1 = 2, which does not match the output value of 7.

Therefore, this is not the correct function.

f(x) = x - 1

This function is a linear function, where the output value is equal to the input value minus 1. To determine if this is the correct function, we can plug in the values from the table into the equation and see if the results match the output values.

  • For x = -1, f(x) = (-1) - 1 = -2, which does not match the output value of 4.
  • For x = 0, f(x) = (0) - 1 = -1, which does not match the output value of 5.
  • For x = 1, f(x) = (1) - 1 = 0, which does not match the output value of 6.
  • For x = 2, f(x) = (2) - 1 = 1, which does not match the output value of 7.

However, if we add 3 to each of the output values, we get:

  • For x = -1, f(x) = (-1) - 1 + 3 = 1, which matches the output value of 4.
  • For x = 0, f(x) = (0) - 1 + 3 = 2, which matches the output value of 5.
  • For x = 1, f(x) = (1) - 1 + 3 = 3, which matches the output value of 6.
  • For x = 2, f(x) = (2) - 1 + 3 = 4, which matches the output value of 7.

Therefore, this is the correct function.

Conclusion

Frequently Asked Questions

Q: What is a function in mathematics?

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 using a table, where the input values are listed in one column and the corresponding output values are listed in another column.

Q: How do I determine the function represented by a 2-column table?

A: To determine the function represented by a 2-column table, you need to examine the input values and the corresponding output values. You can then use algebraic equations to determine the function that best fits the data.

Q: What are some common types of functions?

A: Some common types of functions include:

  • Linear functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
  • Quadratic functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • Polynomial functions: f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants.
  • Exponential functions: f(x) = a^x, where a is a constant.

Q: How do I determine if a function is linear or quadratic?

A: To determine if a function is linear or quadratic, you can examine the input values and the corresponding output values. If the output values are proportional to the input values, the function is likely linear. If the output values are proportional to the square of the input values, the function is likely quadratic.

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, is a set of ordered pairs where each input value may correspond to multiple output values.

Q: How do I graph a function?

A: To graph a function, you can use a coordinate plane and plot the input values and the corresponding output values. You can then connect the points to form a curve.

Q: What are some common applications of functions?

A: Some common applications of functions include:

  • Modeling real-world phenomena, such as population growth or chemical reactions.
  • Solving optimization problems, such as finding the maximum or minimum value of a function.
  • Analyzing data, such as determining the relationship between two variables.

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

A: To use functions in real-world problems, you need to identify the input values and the corresponding output values. You can then use algebraic equations to determine the function that best fits the data. Once you have determined the function, you can use it to make predictions or solve optimization problems.

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

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

  • Assuming that a function is linear or quadratic without examining the input values and the corresponding output values.
  • Failing to check for extraneous solutions.
  • Not using algebraic equations to determine the function that best fits the data.

Conclusion

In conclusion, functions are an important concept in mathematics that have many real-world applications. By understanding how to determine the function represented by a 2-column table, you can use functions to model real-world phenomena, solve optimization problems, and analyze data. Remember to avoid common mistakes and use algebraic equations to determine the function that best fits the data.