Writing the Function in Function Notation

Oct 13, 2025 by recoveryshake.com 132 views

Introduction

In mathematics, a linear function is a function that can be written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. These functions are used to model real-world situations, such as the cost of goods, the distance traveled, and the temperature of a system. In this article, we will explore how to write a linear function represented by the equation $9x + 3y = 12$ using function notation.

The Equation $9x + 3y = 12$

The given equation is $9x + 3y = 12$ . To write this equation in function notation, we need to isolate the variable $y$ . We can do this by subtracting $9x$ from both sides of the equation, which gives us $3y = -9x + 12$ . Next, we can divide both sides of the equation by $3$ , which gives us $y = -3x + 4$ .

Writing the Function in Function Notation

Now that we have isolated the variable $y$ , we can write the function in function notation. The function notation is written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $-3$ and the y-intercept is $4$ . Therefore, the function can be written as $f(x) = -3x + 4$ .

Comparing the Options

Now that we have written the function in function notation, let's compare it with the given options.

Option A: $f(y) = -\frac{1}{3} y + \frac{4}{3}$
Option B: $f(x) = -3x + 4$
Option C: (not provided)

The correct option is Option B: $f(x) = -3x + 4$ . This is because the function is written in function notation with the independent variable $x$ as the input.

Conclusion

In conclusion, the function represented by the equation $9x + 3y = 12$ can be written using function notation as $f(x) = -3x + 4$ . This is a linear function with a slope of $-3$ and a y-intercept of $4$ . We can use this function to model real-world situations, such as the cost of goods, the distance traveled, and the temperature of a system.

Key Takeaways

A linear function can be written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
To write a linear function in function notation, we need to isolate the variable $y$ and then write the function in the form of $f(x) = mx + b$ .
The function notation is written with the independent variable as the input.

Frequently Asked Questions

Q: What is a linear function?

A: A linear function is a function that can be written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do I write a linear function in function notation?

A: To write a linear function in function notation, we need to isolate the variable $y$ and then write the function in the form of $f(x) = mx + b$ .

Q: What is the slope and y-intercept of the function $f(x) = -3x + 4$ ?

A: The slope of the function is $-3$ and the y-intercept is $4$ .

Q: Can I use the function $f(x) = -3x + 4$ to model real-world situations?

Introduction

In our previous article, we explored how to write a linear function represented by the equation $9x + 3y = 12$ using function notation. We also discussed the key takeaways and frequently asked questions related to linear functions. In this article, we will continue to provide more information and answer additional questions related to linear functions.

Q&A Guide

Q: What is the difference between a linear function and a nonlinear function?

A: A linear function is a function that can be written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. A nonlinear function, on the other hand, is a function that cannot be written in this form. Nonlinear functions can have a variety of shapes, including quadratic, cubic, and exponential.

Q: How do I determine if a function is linear or nonlinear?

A: To determine if a function is linear or nonlinear, you can try to write it in the form of $f(x) = mx + b$ . If you can do this, then the function is linear. If you cannot do this, then the function is nonlinear.

Q: What is the slope of a linear function?

A: The slope of a linear function is the coefficient of the $x$ term in the function. For example, in the function $f(x) = -3x + 4$ , the slope is $-3$ .

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

A: The y-intercept of a linear function is the value of the function when $x = 0$ . For example, in the function $f(x) = -3x + 4$ , the y-intercept is $4$ .

Q: Can I use linear functions to model real-world situations?

A: Yes, you can use linear functions to model real-world situations, such as the cost of goods, the distance traveled, and the temperature of a system.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the slope-intercept form of the function, which is $f(x) = mx + b$ . You can also use the point-slope form of the function, which is $f(x) - f(x_0) = m(x - x_0)$ .

Q: What is the equation of a horizontal line?

A: The equation of a horizontal line is $y = b$ , where $b$ is the y-intercept.

Q: What is the equation of a vertical line?

A: The equation of a vertical line is $x = a$ , where $a$ is the x-intercept.

Q: Can I use linear functions to solve systems of equations?

A: Yes, you can use linear functions to solve systems of equations. You can use the method of substitution or the method of elimination to solve the system.

Q: What is the difference between a linear equation and a linear function?

A: A linear equation is an equation that can be written in the form of $ax + by = c$ , where $a$ , $b$ , and $c$ are constants. A linear function, on the other hand, is a function that can be written in the form of $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Conclusion

In conclusion, linear functions are an important concept in mathematics that can be used to model real-world situations. We have provided a Q&A guide to help you understand linear functions better. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of linear functions.