Complete The Table For The Function $f(x) = -2|x|$.$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -3 & 1 \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline \end{tabular} \\]

Completing the Table for the Function $f(x) = -2|x|$

Understanding the Function

The given function $f(x) = -2|x|$ is a linear function that involves the absolute value of $x$. The absolute value function $|x|$ returns the distance of $x$ from zero on the number line. When $x$ is positive, $|x|$ is equal to $x$, and when $x$ is negative, $|x|$ is equal to $-x$. The function $f(x) = -2|x|$ takes this absolute value and multiplies it by $-2$, which means that the function will always return a negative value for any non-zero input.

Completing the Table

To complete the table, we need to find the values of $f(x)$ for $x = -3, -2, -1,$ and $0$. We can do this by plugging in these values into the function $f(x) = -2|x|$.

For $x = -3$

When $x = -3$, the absolute value of $x$ is $|-3| = 3$. Therefore, $f(-3) = -2 \cdot 3 = -6$. However, the table already has $f(-3) = 1$, which is incorrect. We need to correct this.

For $x = -2$

When $x = -2$, the absolute value of $x$ is $|-2| = 2$. Therefore, $f(-2) = -2 \cdot 2 = -4$.

For $x = -1$

When $x = -1$, the absolute value of $x$ is $|-1| = 1$. Therefore, $f(-1) = -2 \cdot 1 = -2$.

For $x = 0$

When $x = 0$, the absolute value of $x$ is $|0| = 0$. Therefore, $f(0) = -2 \cdot 0 = 0$.

Corrected Table

Discussion

The function $f(x) = -2|x|$ is a linear function that always returns a negative value for any non-zero input. The absolute value function $|x|$ is involved in this function, which means that the function will always return a negative value for any non-zero input. The table we completed shows the values of $f(x)$ for $x = -3, -2, -1,$ and $0$. We can see that the function $f(x) = -2|x|$ is a decreasing function, which means that as $x$ increases, $f(x)$ decreases.

Graph of the Function

The graph of the function $f(x) = -2|x|$ is a V-shaped graph that opens downwards. The graph has a minimum point at $(0, 0)$, and it is symmetric about the y-axis. The graph of the function $f(x) = -2|x|$ is a classic example of a linear function that involves the absolute value of $x$.

Properties of the Function

The function $f(x) = -2|x|$ has several properties that make it an interesting function to study. Some of these properties include:

• Domain: The domain of the function $f(x) = -2|x|$ is all real numbers, which means that the function is defined for all values of $x$.
• Range: The range of the function $f(x) = -2|x|$ is all non-positive real numbers, which means that the function always returns a negative value for any non-zero input.
• Symmetry: The function $f(x) = -2|x|$ is symmetric about the y-axis, which means that the graph of the function is the same when reflected about the y-axis.
• Decreasing: The function $f(x) = -2|x|$ is a decreasing function, which means that as $x$ increases, $f(x)$ decreases.

Conclusion

In conclusion, the function $f(x) = -2|x|$ is a linear function that involves the absolute value of $x$. The function always returns a negative value for any non-zero input, and it has several interesting properties that make it an interesting function to study. The table we completed shows the values of $f(x)$ for $x = -3, -2, -1,$ and $0$, and the graph of the function is a V-shaped graph that opens downwards.
Q&A: Completing the Table for the Function $f(x) = -2|x|$

Frequently Asked Questions

We have received several questions from readers regarding the function $f(x) = -2|x|$ and completing the table for this function. Below are some of the most frequently asked questions and their answers.

Q: What is the absolute value function?

A: The absolute value function $|x|$ returns the distance of $x$ from zero on the number line. When $x$ is positive, $|x|$ is equal to $x$, and when $x$ is negative, $|x|$ is equal to $-x$.

Q: Why is the function $f(x) = -2|x|$ always negative for any non-zero input?

A: The function $f(x) = -2|x|$ takes the absolute value of $x$ and multiplies it by $-2$. Since the absolute value of $x$ is always non-negative, multiplying it by $-2$ will always result in a negative value.

Q: What is the domain of the function $f(x) = -2|x|$?

A: The domain of the function $f(x) = -2|x|$ is all real numbers, which means that the function is defined for all values of $x$.

Q: What is the range of the function $f(x) = -2|x|$?

A: The range of the function $f(x) = -2|x|$ is all non-positive real numbers, which means that the function always returns a negative value for any non-zero input.

Q: Is the function $f(x) = -2|x|$ symmetric about the y-axis?

A: Yes, the function $f(x) = -2|x|$ is symmetric about the y-axis, which means that the graph of the function is the same when reflected about the y-axis.

Q: Is the function $f(x) = -2|x|$ a decreasing function?

A: Yes, the function $f(x) = -2|x|$ is a decreasing function, which means that as $x$ increases, $f(x)$ decreases.

Q: How do I complete the table for the function $f(x) = -2|x|$?

A: To complete the table for the function $f(x) = -2|x|$, you need to plug in the values of $x$ into the function and calculate the corresponding values of $f(x)$.

Q: What is the graph of the function $f(x) = -2|x|$?

A: The graph of the function $f(x) = -2|x|$ is a V-shaped graph that opens downwards. The graph has a minimum point at $(0, 0)$, and it is symmetric about the y-axis.

Q: What are some of the properties of the function $f(x) = -2|x|$?

A: Some of the properties of the function $f(x) = -2|x|$ include its domain, range, symmetry, and decreasing nature.

Conclusion

In conclusion, the function $f(x) = -2|x|$ is a linear function that involves the absolute value of $x$. The function always returns a negative value for any non-zero input, and it has several interesting properties that make it an interesting function to study. We hope that this Q&A article has helped to clarify any questions you may have had about the function $f(x) = -2|x|$ and completing the table for this function.