The Function $g(x)=|x-6|-8$ Is Graphed. What Is The Range?A. $\{y \mid Y \ \textgreater \ -8\}$B. $\{y \mid Y \geq -8\}$C. $\{y \mid Y \ \textless \ -8\}$D. $\{y \mid Y \text{ Is All Real Numbers} \}$

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Introduction

In mathematics, the range of a function is the set of all possible output values it can produce for the given input values. Understanding the range of a function is crucial in various mathematical and real-world applications. In this article, we will explore the function g(x) = |x-6|-8 and determine its range.

The Absolute Value Function

The absolute value function |x| is defined as:

|x| = x if x ≥ 0 |x| = -x if x < 0

The absolute value function has a V-shaped graph, with the vertex at (0, 0). The function g(x) = |x-6|-8 is a transformation of the absolute value function. To understand the range of g(x), we need to analyze its graph.

Graphing the Function g(x) = |x-6|-8

To graph the function g(x) = |x-6|-8, we can start by graphing the absolute value function |x-6|. The graph of |x-6| is a V-shaped graph with the vertex at (6, 0). Then, we can shift the graph down by 8 units to obtain the graph of g(x) = |x-6|-8.

Analyzing the Graph

The graph of g(x) = |x-6|-8 consists of two line segments. The first line segment has a slope of 1 and passes through the points (6, -8) and (7, -7). The second line segment has a slope of -1 and passes through the points (6, -8) and (5, -7).

Determining the Range

To determine the range of g(x), we need to find the minimum and maximum values of the function. From the graph, we can see that the minimum value of g(x) is -8, which occurs when x = 6. The maximum value of g(x) is unbounded, as the function increases without bound as x approaches infinity.

Conclusion

Based on the analysis of the graph, we can conclude that the range of g(x) = |x-6|-8 is the set of all real numbers greater than or equal to -8. This is because the minimum value of g(x) is -8, and the function increases without bound as x approaches infinity.

The Final Answer

The final answer is:

B. {y | y ≥ -8}

Additional Information

The range of a function is an important concept in mathematics, and it has various applications in real-world problems. Understanding the range of a function can help us make informed decisions and solve problems more efficiently.

Common Mistakes

When determining the range of a function, it's common to make mistakes such as:

  • Assuming the range is the set of all real numbers, when in fact it's a subset of real numbers.
  • Failing to consider the minimum and maximum values of the function.
  • Not analyzing the graph of the function carefully.

Tips and Tricks

To determine the range of a function, follow these tips and tricks:

  • Analyze the graph of the function carefully.
  • Find the minimum and maximum values of the function.
  • Consider the domain of the function.
  • Use mathematical reasoning and logic to determine the range.

Real-World Applications

Understanding the range of a function has various real-world applications, such as:

  • Modeling population growth and decline.
  • Analyzing financial data and making informed investment decisions.
  • Solving optimization problems in engineering and economics.

Conclusion

Q: What is the range of the function g(x) = |x-6|-8?

A: The range of the function g(x) = |x-6|-8 is the set of all real numbers greater than or equal to -8.

Q: Why is the range of the function g(x) = |x-6|-8 greater than or equal to -8?

A: The range of the function g(x) = |x-6|-8 is greater than or equal to -8 because the minimum value of the function is -8, which occurs when x = 6.

Q: What is the maximum value of the function g(x) = |x-6|-8?

A: The maximum value of the function g(x) = |x-6|-8 is unbounded, as the function increases without bound as x approaches infinity.

Q: How can I determine the range of a function?

A: To determine the range of a function, you can analyze the graph of the function, find the minimum and maximum values of the function, and consider the domain of the function.

Q: What are some common mistakes to avoid when determining the range of a function?

A: Some common mistakes to avoid when determining the range of a function include:

  • Assuming the range is the set of all real numbers, when in fact it's a subset of real numbers.
  • Failing to consider the minimum and maximum values of the function.
  • Not analyzing the graph of the function carefully.

Q: What are some tips and tricks for determining the range of a function?

A: Some tips and tricks for determining the range of a function include:

  • Analyzing the graph of the function carefully.
  • Finding the minimum and maximum values of the function.
  • Considering the domain of the function.
  • Using mathematical reasoning and logic to determine the range.

Q: What are some real-world applications of understanding the range of a function?

A: Some real-world applications of understanding the range of a function include:

  • Modeling population growth and decline.
  • Analyzing financial data and making informed investment decisions.
  • Solving optimization problems in engineering and economics.

Q: Can you provide an example of how to determine the range of a function?

A: Let's consider the function f(x) = 2x - 5. To determine the range of this function, we can analyze the graph of the function, find the minimum and maximum values of the function, and consider the domain of the function.

The graph of the function f(x) = 2x - 5 is a straight line with a slope of 2 and a y-intercept of -5. The minimum value of the function is -5, which occurs when x = 0. The maximum value of the function is unbounded, as the function increases without bound as x approaches infinity.

Therefore, the range of the function f(x) = 2x - 5 is the set of all real numbers greater than or equal to -5.

Q: Can you provide an example of how to use the range of a function in a real-world application?

A: Let's consider a company that is planning to invest in a new project. The company wants to determine the maximum return on investment (ROI) for the project.

To determine the maximum ROI, the company can use the range of the function f(x) = 2x - 5, where x is the amount of money invested in the project. The range of the function is the set of all real numbers greater than or equal to -5.

The company can use this information to determine the maximum ROI for the project by analyzing the graph of the function, finding the minimum and maximum values of the function, and considering the domain of the function.

By using the range of the function, the company can make informed decisions about the investment and maximize the ROI for the project.

Conclusion

In conclusion, the range of the function g(x) = |x-6|-8 is the set of all real numbers greater than or equal to -8. Understanding the range of a function is crucial in various mathematical and real-world applications. By analyzing the graph of the function, finding the minimum and maximum values of the function, and considering the domain of the function, we can determine the range of a function and make informed decisions.