The Table Represents The Function $f(x$\].$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -4 & -66 \\ \hline -3 & -29 \\ \hline -2 & -10 \\ \hline -1 & -3 \\ \hline 0 & -2 \\ \hline 1 & -1 \\ \hline 2 & 6

$$

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. The table represents the function $f(x)$, where the input $x$ is related to the output $f(x)$. In this article, we will analyze the given table and determine the function $f(x)$.

The Table

$x$	$f(x)$
-4	-66
-3	-29
-2	-10
-1	-3
0	-2
1	-1
2	6

Observations

From the table, we can observe the following:

• The function $f(x)$ is defined for all integer values of $x$ from -4 to 2.
• The function $f(x)$ is decreasing for $x$ values from -4 to 0.
• The function $f(x)$ is increasing for $x$ values from 0 to 2.
• The function $f(x)$ has a minimum value of -66 at $x$ = -4.
• The function $f(x)$ has a maximum value of 6 at $x$ = 2.

Determining the Function

To determine the function $f(x)$, we need to find a mathematical expression that satisfies the given table. Let's start by analyzing the differences between consecutive values of $f(x)$.

$x$	$f(x)$	$\Delta f(x)$
-4	-66	-
-3	-29	37
-2	-10	19
-1	-3	7
0	-2	1
1	-1	1
2	6	7

From the table, we can see that the differences between consecutive values of $f(x)$ are:

• $\Delta f(-4) = -$
• $\Delta f(-3) = 37$
• $\Delta f(-2) = 19$
• $\Delta f(-1) = 7$
• $\Delta f(0) = 1$
• $\Delta f(1) = 1$
• $\Delta f(2) = 7$

The differences between consecutive values of $f(x)$ are not constant, but they are increasing and decreasing in a pattern. This suggests that the function $f(x)$ may be a quadratic function.

Quadratic Function

A quadratic function has the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. To determine the function $f(x)$, we need to find the values of $a$, $b$, and $c$ that satisfy the given table.

Let's start by assuming that the function $f(x)$ is a quadratic function of the form $f(x) = ax^2 + bx + c$. We can then use the given table to find the values of $a$, $b$, and $c$.

Using the values of $f(x)$ and $x$ from the table, we can write the following equations:

• $-66 = a(-4)^2 + b(-4) + c$
• $-29 = a(-3)^2 + b(-3) + c$
• $-10 = a(-2)^2 + b(-2) + c$
• $-3 = a(-1)^2 + b(-1) + c$
• $-2 = a(0)^2 + b(0) + c$
• $-1 = a(1)^2 + b(1) + c$
• $6 = a(2)^2 + b(2) + c$

Solving these equations simultaneously, we get:

• $a = -3$
• $b = 4$
• $c = -2$

Therefore, the function $f(x)$ is given by:

$f(x) = -3x^2 + 4x - 2$

Conclusion

In this article, we analyzed the given table and determined the function $f(x)$. We observed that the function $f(x)$ is a quadratic function of the form $f(x) = -3x^2 + 4x - 2$. The function $f(x)$ has a minimum value of -66 at $x$ = -4 and a maximum value of 6 at $x$ = 2. The function $f(x)$ is decreasing for $x$ values from -4 to 0 and increasing for $x$ values from 0 to 2.

References

• [1] "Functions" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference
• [3] "Table of Functions" by Wolfram Alpha

Further Reading

• "Functions and Relations" by MIT OpenCourseWare
• "Quadratic Equations and Functions" by Purplemath
• "Table of Functions" by Mathway
  Q&A: The Table Represents the Function $f(x)$ =====================================================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the table represents the function $f(x)$.

Q: What is the function $f(x)$?

A: The function $f(x)$ is a quadratic function of the form $f(x) = -3x^2 + 4x - 2$.

Q: What is the domain of the function $f(x)$?

A: The domain of the function $f(x)$ is all integer values of $x$ from -4 to 2.

Q: What is the range of the function $f(x)$?

A: The range of the function $f(x)$ is all real values of $f(x)$ from -66 to 6.

Q: Is the function $f(x)$ increasing or decreasing?

A: The function $f(x)$ is decreasing for $x$ values from -4 to 0 and increasing for $x$ values from 0 to 2.

Q: What is the minimum value of the function $f(x)$?

A: The minimum value of the function $f(x)$ is -66, which occurs at $x$ = -4.

Q: What is the maximum value of the function $f(x)$?

A: The maximum value of the function $f(x)$ is 6, which occurs at $x$ = 2.

Q: How can I graph the function $f(x)$?

A: You can graph the function $f(x)$ by using a graphing calculator or a computer algebra system.

Q: Can I use the function $f(x)$ to model real-world phenomena?

A: Yes, the function $f(x)$ can be used to model real-world phenomena such as the motion of an object under the influence of gravity.

Q: How can I find the values of $a$, $b$, and $c$ for a quadratic function?

A: You can find the values of $a$, $b$, and $c$ for a quadratic function by using the given values of $f(x)$ and $x$ to write a system of equations and then solving the system.

Q: What is the significance of the quadratic function $f(x)$?

A: The quadratic function $f(x)$ is significant because it can be used to model a wide range of real-world phenomena, including the motion of objects under the influence of gravity.

Common Misconceptions

• Misconception: The function $f(x)$ is a linear function.

• Reality: The function $f(x)$ is a quadratic function of the form $f(x) = -3x^2 + 4x - 2$.

• Misconception: The function $f(x)$ is always increasing.

• Reality: The function $f(x)$ is decreasing for $x$ values from -4 to 0 and increasing for $x$ values from 0 to 2.

• Misconception: The function $f(x)$ has a maximum value of 10.

• Reality: The function $f(x)$ has a maximum value of 6, which occurs at $x$ = 2.

Conclusion

In this article, we answered some frequently asked questions about the table represents the function $f(x)$. We also discussed some common misconceptions about the function $f(x)$. We hope that this article has been helpful in clarifying any confusion about the function $f(x)$.

References

• [1] "Functions" by Khan Academy
• [2] "Quadratic Functions" by Math Open Reference
• [3] "Table of Functions" by Wolfram Alpha

Further Reading

• "Functions and Relations" by MIT OpenCourseWare
• "Quadratic Equations and Functions" by Purplemath
• "Table of Functions" by Mathway