Which Function Is Linear?A. F ( X ) = 8 X 2 + 2 F(x) = 8x^2 + 2 F ( X ) = 8 X 2 + 2 B. F ( X ) = 8 X + 2 F(x) = \frac{8}{x} + 2 F ( X ) = X 8 ​ + 2 C. F ( X ) = 8 X 2 − 2 F(x) = 8x^2 - 2 F ( X ) = 8 X 2 − 2 D. F ( X ) = X 8 − 2 F(x) = \frac{x}{8} - 2 F ( X ) = 8 X ​ − 2

Introduction

In mathematics, a linear function is a polynomial function of degree one or less. It is a function that can be written in the form of f(x) = mx + b, where m and b are constants. Linear functions are an essential concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will discuss the characteristics of linear functions and identify which of the given functions is linear.

Characteristics of Linear Functions

A linear function has several key characteristics:

• Constant Rate of Change: The rate of change of a linear function is constant. This means that for every unit increase in the input (x), the output (f(x)) increases by the same amount.
• Straight Line Graph: The graph of a linear function is a straight line. This is because the function can be written in the form of f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
• No Higher-Degree Terms: A linear function has no terms with a degree higher than one. This means that the function cannot have any squared, cubed, or higher-degree terms.

Examples of Linear Functions

Here are some examples of linear functions:

• f(x) = 2x + 3
• f(x) = -4x + 2
• f(x) = x - 1
• f(x) = 2x^2 + 3 (This is not a linear function because it has a squared term)

Which Function is Linear?

Now that we have discussed the characteristics of linear functions, let's examine the given options:

A. $f(x) = 8x^2 + 2$

This function is not linear because it has a squared term.

B. $f(x) = \frac{8}{x} + 2$

This function is not linear because it has a fraction with a variable in the denominator.

C. $f(x) = 8x^2 - 2$

This function is not linear because it has a squared term.

D. $f(x) = \frac{x}{8} - 2$

This function is linear because it can be written in the form of f(x) = mx + b, where m = 1/8 and b = -2.

Conclusion

In conclusion, a linear function is a polynomial function of degree one or less. It has a constant rate of change, a straight line graph, and no higher-degree terms. We have examined the given options and determined that only option D is a linear function.

Key Takeaways

• A linear function is a polynomial function of degree one or less.
• A linear function has a constant rate of change.
• A linear function has a straight line graph.
• A linear function has no higher-degree terms.
• Option D is the only linear function among the given options.

Additional Resources

For more information on linear functions, we recommend the following resources:

• Khan Academy: Linear Functions
• Math Is Fun: Linear Functions
• Wolfram MathWorld: Linear Function

Frequently Asked Questions

Q: What is a linear function? A: A linear function is a polynomial function of degree one or less.

Q: What are the characteristics of a linear function? A: A linear function has a constant rate of change, a straight line graph, and no higher-degree terms.

Q: Which of the given options is a linear function? A: Option D is the only linear function among the given options.

Introduction

In our previous article, we discussed the characteristics of linear functions and identified which of the given functions is linear. In this article, we will provide a comprehensive Q&A guide to help you better understand linear functions.

Q&A

Q: What is a linear function?

A: A linear function is a polynomial function of degree one or less. It can be written in the form of f(x) = mx + b, where m and b are constants.

Q: What are the characteristics of a linear function?

A: A linear function has a constant rate of change, a straight line graph, and no higher-degree terms.

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

A: To determine if a function is linear, check if it can be written in the form of f(x) = mx + b, where m and b are constants. If it can be written in this form, then it is a linear function.

Q: What are some examples of linear functions?

A: Some examples of linear functions include:

• f(x) = 2x + 3
• f(x) = -4x + 2
• f(x) = x - 1
• f(x) = 2x^2 + 3 (This is not a linear function because it has a squared term)

Q: Can a linear function have a squared term?

A: No, a linear function cannot have a squared term. If a function has a squared term, then it is not a linear function.

Q: Can a linear function have a fraction with a variable in the denominator?

A: No, a linear function cannot have a fraction with a variable in the denominator. If a function has a fraction with a variable in the denominator, then it is not a linear function.

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

A: A linear function is a polynomial function of degree one or less, while a quadratic function is a polynomial function of degree two. A linear function has a straight line graph, while a quadratic function has a parabolic graph.

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

A: Yes, linear functions can be used to model real-world situations. For example, a linear function can be used to model the cost of producing a product, the distance traveled by an object, or the temperature of a substance.

Q: How do I graph a linear function?

A: To graph a linear function, use the slope-intercept form of the function, which is f(x) = mx + b. The slope of the line is m, and the y-intercept is b.

Q: Can a linear function have a negative slope?

A: Yes, a linear function can have a negative slope. A negative slope indicates that the function is decreasing as x increases.

Q: Can a linear function have a positive slope?

A: Yes, a linear function can have a positive slope. A positive slope indicates that the function is increasing as x increases.

Q: Can a linear function have a zero slope?

A: Yes, a linear function can have a zero slope. A zero slope indicates that the function is a horizontal line.

Q: Can a linear function have a vertical asymptote?

A: No, a linear function cannot have a vertical asymptote. A vertical asymptote occurs when a function approaches infinity as x approaches a certain value.

Q: Can a linear function have a horizontal asymptote?

A: Yes, a linear function can have a horizontal asymptote. A horizontal asymptote occurs when a function approaches a certain value as x approaches infinity.

Conclusion

In conclusion, linear functions are an essential concept in mathematics, and they have numerous applications in various fields. We hope that this Q&A guide has helped you better understand linear functions and how to identify them.

Key Takeaways

• A linear function is a polynomial function of degree one or less.
• A linear function has a constant rate of change, a straight line graph, and no higher-degree terms.
• A linear function can be written in the form of f(x) = mx + b, where m and b are constants.
• A linear function can have a negative slope, a positive slope, or a zero slope.
• A linear function can have a horizontal asymptote but not a vertical asymptote.

Additional Resources

For more information on linear functions, we recommend the following resources:

• Khan Academy: Linear Functions
• Math Is Fun: Linear Functions
• Wolfram MathWorld: Linear Function

Frequently Asked Questions

Q: What is a linear function? A: A linear function is a polynomial function of degree one or less.

Q: What are the characteristics of a linear function? A: A linear function has a constant rate of change, a straight line graph, and no higher-degree terms.

Q: How do I determine if a function is linear? A: To determine if a function is linear, check if it can be written in the form of f(x) = mx + b, where m and b are constants.

Q: What are some examples of linear functions? A: Some examples of linear functions include f(x) = 2x + 3, f(x) = -4x + 2, and f(x) = x - 1.

Q: Can a linear function have a squared term? A: No, a linear function cannot have a squared term.

Q: Can a linear function have a fraction with a variable in the denominator? A: No, a linear function cannot have a fraction with a variable in the denominator.