Given The Following Linear Function, Sketch The Graph Of The Function And Find The Domain And Range. F ( X ) = 2 3 X − 3 F(x)=\frac{2}{3} X-3 F ( X ) = 3 2 ​ X − 3

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Introduction

Linear functions are a fundamental concept in mathematics, and understanding their graphing and domain/range analysis is crucial for solving various mathematical problems. In this article, we will explore the graphing and domain/range analysis of the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3. We will also discuss the importance of linear functions in real-world applications.

What is a Linear Function?

A linear function is a polynomial function of degree one, which means it can be written in the form f(x)=ax+bf(x)=ax+b, where aa and bb are constants, and xx is the variable. Linear functions have a constant rate of change, and their graphs are straight lines.

Graphing the Linear Function

To graph the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3, we need to find two points on the line. We can do this by substituting different values of xx into the function and finding the corresponding values of yy.

Let's start by finding the yy-intercept, which is the point where the line intersects the yy-axis. To find the yy-intercept, we set x=0x=0 and solve for yy:

f(0)=23(0)−3=−3f(0)=\frac{2}{3} (0)-3=-3

So, the yy-intercept is (−3,0)(-3,0).

Next, let's find another point on the line. We can do this by substituting a value of xx into the function and finding the corresponding value of yy. Let's choose x=3x=3:

f(3)=23(3)−3=1f(3)=\frac{2}{3} (3)-3=1

So, the point (3,1)(3,1) is on the line.

Now that we have two points on the line, we can graph the line by drawing a straight line through the points (−3,0)(-3,0) and (3,1)(3,1).

Domain and Range Analysis

The domain of a function is the set of all possible input values for which the function is defined. In the case of a linear function, the domain is all real numbers, which means that the function is defined for all values of xx.

The range of a function is the set of all possible output values for which the function is defined. In the case of a linear function, the range is also all real numbers, which means that the function can take on any value.

To find the domain and range of the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3, we can use the following rules:

  • The domain of a linear function is all real numbers.
  • The range of a linear function is all real numbers.

Therefore, the domain and range of the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3 are both all real numbers.

Real-World Applications

Linear functions have many real-world applications, including:

  • Physics: Linear functions are used to describe the motion of objects under constant acceleration.
  • Economics: Linear functions are used to model the relationship between two variables, such as the price of a good and the quantity demanded.
  • Computer Science: Linear functions are used in algorithms for solving linear systems of equations.

Conclusion

In this article, we have explored the graphing and domain/range analysis of the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3. We have also discussed the importance of linear functions in real-world applications. Linear functions are a fundamental concept in mathematics, and understanding their graphing and domain/range analysis is crucial for solving various mathematical problems.

References

Glossary

  • Linear Function: A polynomial function of degree one, which can be written in the form f(x)=ax+bf(x)=ax+b, where aa and bb are constants, and xx is the variable.
  • Domain: The set of all possible input values for which the function is defined.
  • Range: The set of all possible output values for which the function is defined.
  • Graph: A visual representation of a function, which shows the relationship between the input and output values.
    Linear Functions: Q&A =========================

Introduction

In our previous article, we explored the graphing and domain/range analysis of the linear function f(x)=23x−3f(x)=\frac{2}{3} x-3. In this article, we will answer some frequently asked questions about linear functions.

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

A: A linear function is a polynomial function of degree one, which means it can be written in the form f(x)=ax+bf(x)=ax+b, where aa and bb are constants, and xx is the variable. A non-linear function, on the other hand, is a polynomial function of degree greater than one, which means it cannot be written in the form f(x)=ax+bf(x)=ax+b.

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

A: To determine if a function is linear or non-linear, you can try to write it in the form f(x)=ax+bf(x)=ax+b. If you can write the function in this form, then it is a linear function. If you cannot write the function in this form, then it is a non-linear function.

Q: What is the slope of a linear function?

A: The slope of a linear function is the coefficient of the variable xx, which is denoted by aa. The slope represents the rate of change of the function.

Q: How do I find the slope of a linear function?

A: To find the slope of a linear function, you can use the formula m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}, where (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2) are two points on the line.

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

A: The y-intercept of a linear function is the point where the line intersects the y-axis. It is denoted by the symbol (0,b)(0,b), where bb is the constant term in the function.

Q: How do I find the y-intercept of a linear function?

A: To find the y-intercept of a linear function, you can set x=0x=0 and solve for yy. This will give you the point where the line intersects the y-axis.

Q: What is the domain and range of a linear function?

A: The domain of a linear function is all real numbers, which means that the function is defined for all values of xx. The range of a linear function is also all real numbers, which means that the function can take on any value.

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)=ax+bf(x)=ax+b. You can also use the two-point form of the function, which is f(x)=y2−y1x2−x1(x−x1)+y1f(x)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1.

Q: What are some real-world applications of linear functions?

A: Linear functions have many real-world applications, including:

  • Physics: Linear functions are used to describe the motion of objects under constant acceleration.
  • Economics: Linear functions are used to model the relationship between two variables, such as the price of a good and the quantity demanded.
  • Computer Science: Linear functions are used in algorithms for solving linear systems of equations.

Conclusion

In this article, we have answered some frequently asked questions about linear functions. We hope that this article has been helpful in clarifying some of the concepts related to linear functions.

References

Glossary

  • Linear Function: A polynomial function of degree one, which can be written in the form f(x)=ax+bf(x)=ax+b, where aa and bb are constants, and xx is the variable.
  • Domain: The set of all possible input values for which the function is defined.
  • Range: The set of all possible output values for which the function is defined.
  • Graph: A visual representation of a function, which shows the relationship between the input and output values.
  • Slope: The coefficient of the variable xx in a linear function, which represents the rate of change of the function.
  • Y-intercept: The point where the line intersects the y-axis, which is denoted by the symbol (0,b)(0,b).