Complete The Table.$\[ \begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{$f(x)=|-2x|$} \\ \hline $x$ & $f(x)$ \\ \hline -1 & $\square$ \\ \hline 0 & $\square$ \\ \hline 1 & $\square$ \\ \hline 2 & $\square$

Introduction to the Function

The given function is $f(x) = |-2x|$. This function involves the absolute value of $-2x$, which means that the output will always be non-negative. The absolute value function is defined as $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$. In this case, we have $|-2x| = -(-2x)$ if $-2x \geq 0$ and $|-2x| = -(-2x)$ if $-2x < 0$. However, since the output of the function is always non-negative, we can simplify the function to $f(x) = 2x$ if $x \geq 0$ and $f(x) = 2x$ if $x < 0$. This is because the absolute value function will always return a non-negative value.

Understanding the Table

The table provided has four rows, each representing a different input value of $x$. The columns represent the input value $x$ and the corresponding output value $f(x)$. The table is incomplete, with the output values represented by $\square$.

Completing the Table

To complete the table, we need to find the output values for each input value of $x$. We can do this by plugging in the values of $x$ into the function $f(x) = |-2x|$.

Input Value $x = -1$

For $x = -1$, we have $f(-1) = |-2(-1)| = |-2| = 2$. Therefore, the output value for $x = -1$ is $2$.

Input Value $x = 0$

For $x = 0$, we have $f(0) = |-2(0)| = |0| = 0$. Therefore, the output value for $x = 0$ is $0$.

Input Value $x = 1$

For $x = 1$, we have $f(1) = |-2(1)| = |-2| = 2$. Therefore, the output value for $x = 1$ is $2$.

Input Value $x = 2$

For $x = 2$, we have $f(2) = |-2(2)| = |-4| = 4$. Therefore, the output value for $x = 2$ is $4$.

Completed Table

The completed table is as follows:

Discussion and Conclusion

In this article, we have completed the table for the given function $f(x) = |-2x|$. We have found the output values for each input value of $x$ by plugging in the values into the function. The completed table shows that the output values are non-negative, as expected. This is because the absolute value function always returns a non-negative value. The completed table can be used to visualize the behavior of the function and to understand its properties.

Key Takeaways

• The function $f(x) = |-2x|$ involves the absolute value of $-2x$.
• The output of the function is always non-negative.
• The completed table shows that the output values are non-negative.
• The function can be simplified to $f(x) = 2x$ if $x \geq 0$ and $f(x) = 2x$ if $x < 0$.

Future Directions

In future articles, we can explore other properties of the function, such as its domain and range, and its behavior under different transformations. We can also use the completed table to visualize the behavior of the function and to understand its properties in more detail.

Introduction

In our previous article, we completed the table for the given function $f(x) = |-2x|$. In this article, we will answer some frequently asked questions (FAQs) about the function.

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

A: The domain of the function is all real numbers, i.e., $x \in \mathbb{R}$. This is because the absolute value function is defined for all real numbers.

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

A: The range of the function is all non-negative real numbers, i.e., $f(x) \geq 0$. This is because the absolute value function always returns a non-negative value.

Q: How can we simplify the function $f(x) = |-2x|$?

A: We can simplify the function to $f(x) = 2x$ if $x \geq 0$ and $f(x) = 2x$ if $x < 0$. This is because the absolute value function will always return a non-negative value.

Q: What is the behavior of the function $f(x) = |-2x|$ under different transformations?

A: The function $f(x) = |-2x|$ is a linear function, and its behavior under different transformations is as follows:

• If we multiply the function by a constant $c$, the new function is $f(x) = c|-2x|$.
• If we add a constant $k$ to the function, the new function is $f(x) = |-2x| + k$.
• If we replace $x$ with $-x$, the new function is $f(x) = |-2(-x)| = |-2x|$.

Q: Can we use the function $f(x) = |-2x|$ to model real-world phenomena?

A: Yes, we can use the function $f(x) = |-2x|$ to model real-world phenomena that involve absolute value, such as:

• The distance between two points on a number line.
• The magnitude of a vector.
• The absolute temperature.

Q: How can we visualize the behavior of the function $f(x) = |-2x|$?

A: We can visualize the behavior of the function by plotting its graph. The graph of the function is a V-shaped graph that opens upwards, with the vertex at the origin (0, 0).

Q: Can we use the function $f(x) = |-2x|$ to solve equations and inequalities?

A: Yes, we can use the function $f(x) = |-2x|$ to solve equations and inequalities that involve absolute value, such as:

• Solving equations of the form $|ax| = b$.
• Solving inequalities of the form $|ax| > b$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about the function $f(x) = |-2x|$. We have discussed the domain and range of the function, its behavior under different transformations, and its use in modeling real-world phenomena. We have also provided some tips on how to visualize the behavior of the function and how to use it to solve equations and inequalities.