The Table Represents The Function $f(x)$. \[ \begin{tabular}{|c|c|} \hline X$ & F ( X ) F(x) F ( X ) \ \hline -3 & -9 \ \hline -2 & -6 \ \hline -1 & -3 \ \hline 0 & 0 \ \hline 1 & 3 \ \hline 2 & 6 \ \hline 3 & 9

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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. The table represents the function $f(x)$, where the input $x$ is related to the output $f(x)$. In this article, we will explore the relationship between the input and output values in the table and determine the function that represents this relationship.

Analyzing the Table

The table provides a set of input-output pairs, where each input value $x$ is associated with a corresponding output value $f(x)$. By examining the table, we can identify patterns and relationships between the input and output values.

$x$	$f(x)$
-3	-9
-2	-6
-1	-3
0	0
1	3
2	6
3	9

Identifying the Pattern

Upon closer inspection, we can observe that the output values $f(x)$ are increasing by 3 for each consecutive input value $x$. This suggests that the function $f(x)$ is a linear function, where the output value increases by a constant rate for each unit increase in the input value.

Determining the Function

To determine the function that represents this relationship, we can use the input-output pairs in the table to find the slope and y-intercept of the linear function. The slope represents the rate of change of the output value with respect to the input value, while the y-intercept represents the value of the output when the input is zero.

Let's examine the input-output pairs in the table:

• For $x = -3$, $f(x) = -9$
• For $x = -2$, $f(x) = -6$
• For $x = -1$, $f(x) = -3$
• For $x = 0$, $f(x) = 0$
• For $x = 1$, $f(x) = 3$
• For $x = 2$, $f(x) = 6$
• For $x = 3$, $f(x) = 9$

We can see that the output value increases by 3 for each consecutive input value. This suggests that the slope of the linear function is 3.

To find the y-intercept, we can use the input-output pair $(0, 0)$, where the input is zero and the output is also zero. This suggests that the y-intercept is 0.

Therefore, the function that represents this relationship is:

$f(x) = 3x$

Conclusion

In this article, we analyzed the table that represents the function $f(x)$ and determined the function that represents this relationship. We identified the pattern of increasing output values for consecutive input values and used the input-output pairs in the table to find the slope and y-intercept of the linear function. The function that represents this relationship is $f(x) = 3x$, where the output value increases by 3 for each unit increase in the input value.

Understanding the Graph of the Function

The graph of the function $f(x) = 3x$ is a straight line that passes through the origin $(0, 0)$. The slope of the line is 3, which represents the rate of change of the output value with respect to the input value.

To visualize the graph of the function, we can plot the input-output pairs in the table on a coordinate plane.

$x$	$f(x)$
-3	-9
-2	-6
-1	-3
0	0
1	3
2	6
3	9

The graph of the function is a straight line that passes through the points $(-3, -9)$, $(-2, -6)$, $(-1, -3)$, $(0, 0)$, $(1, 3)$, $(2, 6)$, and $(3, 9)$.

Properties of the Function

The function $f(x) = 3x$ has several important properties that are worth noting:

• Domain: The domain of the function is all real numbers, denoted by $(-\infty, \infty)$.
• Range: The range of the function is all real numbers, denoted by $(-\infty, \infty)$.
• Slope: The slope of the function is 3, which represents the rate of change of the output value with respect to the input value.
• Y-intercept: The y-intercept of the function is 0, which represents the value of the output when the input is zero.

Real-World Applications

The function $f(x) = 3x$ has several real-world applications, including:

• Cost and revenue analysis: The function can be used to model the cost and revenue of a business, where the input represents the number of units produced or sold and the output represents the total cost or revenue.
• Population growth: The function can be used to model the growth of a population, where the input represents the time and the output represents the population size.
• Finance: The function can be used to model the growth of an investment, where the input represents the time and the output represents the investment value.

Conclusion

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the function $f(x) = 3x$. Whether you are a student, a teacher, or simply someone interested in mathematics, this article will provide you with a deeper understanding of the function and its properties.

Q: What is the domain of the function $f(x) = 3x$?

A: The domain of the function $f(x) = 3x$ is all real numbers, denoted by $(-\infty, \infty)$. This means that the function can take on any real value as input.

Q: What is the range of the function $f(x) = 3x$?

A: The range of the function $f(x) = 3x$ is also all real numbers, denoted by $(-\infty, \infty)$. This means that the function can take on any real value as output.

Q: What is the slope of the function $f(x) = 3x$?

A: The slope of the function $f(x) = 3x$ is 3. This means that for every unit increase in the input value, the output value increases by 3.

Q: What is the y-intercept of the function $f(x) = 3x$?

A: The y-intercept of the function $f(x) = 3x$ is 0. This means that when the input value is 0, the output value is also 0.

Q: How do I graph the function $f(x) = 3x$?

A: To graph the function $f(x) = 3x$, you can use a coordinate plane and plot the input-output pairs in the table. The graph will be a straight line that passes through the origin $(0, 0)$.

Q: What are some real-world applications of the function $f(x) = 3x$?

A: The function $f(x) = 3x$ has several real-world applications, including:

• Cost and revenue analysis: The function can be used to model the cost and revenue of a business, where the input represents the number of units produced or sold and the output represents the total cost or revenue.
• Population growth: The function can be used to model the growth of a population, where the input represents the time and the output represents the population size.
• Finance: The function can be used to model the growth of an investment, where the input represents the time and the output represents the investment value.

Q: How do I use the function $f(x) = 3x$ to solve problems?

A: To use the function $f(x) = 3x$ to solve problems, you can substitute the input value into the function and evaluate the output value. For example, if you want to find the output value when the input value is 4, you can substitute 4 into the function and evaluate the output value: $f(4) = 3(4) = 12$.

Q: What are some common mistakes to avoid when working with the function $f(x) = 3x$?

A: Some common mistakes to avoid when working with the function $f(x) = 3x$ include:

• Not checking the domain and range: Make sure to check the domain and range of the function before using it to solve problems.
• Not evaluating the function correctly: Make sure to evaluate the function correctly by substituting the input value into the function and following the order of operations.
• Not considering the y-intercept: Make sure to consider the y-intercept of the function when graphing it or using it to solve problems.

Conclusion

In conclusion, the function $f(x) = 3x$ is a simple yet powerful function that has many real-world applications. By understanding the properties of the function, including its domain, range, slope, and y-intercept, you can use it to solve problems and model real-world situations. We hope this Q&A article has been helpful in answering your questions and providing you with a deeper understanding of the function.