Determining the Function Equation

Oct 13, 2025 by magicalceylontours.com 129 views

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 a way of describing a relationship between two variables, where each input corresponds to exactly one output. In this article, we will explore the function represented by the given table and understand its properties.

The Table

The table given represents a function, where the input values are represented by the variable $x$ and the corresponding output values are represented by the variable $f(x)$ . The table has three ordered pairs: $(2, 6)$ , $(7, 3)$ , and $(9, 5)$ .

$x$	$f(x)$
2	6
7	3
9	5

Analyzing the Function

To understand the function represented by the table, we need to analyze the ordered pairs. We can see that each input value corresponds to exactly one output value. For example, when $x = 2$ , the output value is $f(x) = 6$ . Similarly, when $x = 7$ , the output value is $f(x) = 3$ .

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. From the table, we can see that the domain of the function is $\{2, 7, 9\}$ , and the range is $\{3, 5, 6\}$ .

Determining the Function

To determine the function represented by the table, we need to find a rule that describes the relationship between the input and output values. One way to do this is to look for a pattern in the ordered pairs. In this case, we can see that the output value is increasing as the input value increases.

Finding the Rule

Let's examine the ordered pairs more closely. We can see that when $x = 2$ , the output value is $f(x) = 6$ . When $x = 7$ , the output value is $f(x) = 3$ . When $x = 9$ , the output value is $f(x) = 5$ . We can see that the output value is decreasing as the input value increases.

Determining the Function Equation

Based on the pattern we observed, we can determine the function equation. Let's assume that the function is a linear function, which means that the output value is proportional to the input value. We can write the function equation as $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Finding the Slope

To find the slope, we can use the formula $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$ . Let's use the ordered pairs $(2, 6)$ and $(7, 3)$ to find the slope.

$m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{3 - 6}{7 - 2} = \frac{-3}{5} = -0.6$

Finding the Y-Intercept

To find the y-intercept, we can use the formula $b = f(x) - mx$ . Let's use the ordered pair $(2, 6)$ to find the y-intercept.

$b = f(x) - mx = 6 - (-0.6)(2) = 6 + 1.2 = 7.2$

Determining the Function Equation

Now that we have found the slope and y-intercept, we can determine the function equation.

$f(x) = mx + b = -0.6x + 7.2$

Conclusion

In this article, we explored the function represented by the given table and understood its properties. We analyzed the ordered pairs, determined the domain and range, and found the function equation. We also found the slope and y-intercept, which helped us determine the function equation.

Final Thoughts

The function represented by the table is a linear function, which means that the output value is proportional to the input value. The function equation is $f(x) = -0.6x + 7.2$ . This function can be used to model a variety of real-world situations, such as the cost of a product based on the quantity purchased.

References

[1] "Functions" by Khan Academy
[2] "Linear Functions" by Math Is Fun
[3] "Functions and Relations" by Purplemath

Additional Resources

[1] "Functions" by Wolfram MathWorld
[2] "Linear Functions" by IXL
[3] "Functions and Relations" by Math Open Reference
Q&A: Understanding the Function Represented by the Table =====================================================

Introduction

In our previous article, we explored the function represented by the given table and understood its properties. We analyzed the ordered pairs, determined the domain and range, and found the function equation. In this article, we will answer some frequently asked questions about the function represented by the table.

Q: What is the domain of the function?

A: The domain of the function is the set of all possible input values. From the table, we can see that the domain of the function is $\{2, 7, 9\}$ .

Q: What is the range of the function?

A: The range of the function is the set of all possible output values. From the table, we can see that the range of the function is $\{3, 5, 6\}$ .

Q: What is the function equation?

A: The function equation is $f(x) = -0.6x + 7.2$ . This equation describes the relationship between the input and output values of the function.

Q: Is the function linear?

A: Yes, the function is linear. This means that the output value is proportional to the input value.

Q: Can the function be used to model real-world situations?

A: Yes, the function can be used to model a variety of real-world situations, such as the cost of a product based on the quantity purchased.

Q: How can I determine the function equation?

A: To determine the function equation, you can analyze the ordered pairs and look for a pattern. You can also use the formula $f(x) = mx + b$ to find the slope and y-intercept.

Q: What is the slope of the function?

A: The slope of the function is $-0.6$ . This means that for every unit increase in the input value, the output value decreases by $0.6$ units.

Q: What is the y-intercept of the function?

A: The y-intercept of the function is $7.2$ . This means that when the input value is $0$ , the output value is $7.2$ .

Q: Can the function be used to model situations where the output value is not proportional to the input value?

A: No, the function cannot be used to model situations where the output value is not proportional to the input value. The function is linear, which means that the output value is proportional to the input value.

Q: How can I use the function to model real-world situations?

A: To use the function to model real-world situations, you can substitute the input values into the function equation and solve for the output values. For example, if you want to find the cost of a product based on the quantity purchased, you can substitute the quantity into the function equation and solve for the cost.

Conclusion

In this article, we answered some frequently asked questions about the function represented by the table. We discussed the domain and range of the function, the function equation, and how to use the function to model real-world situations.