What Is The Range Of The Function G ( X ) = ∣ X − 12 ∣ − 2 G(x) = |x - 12| - 2 G ( X ) = ∣ X − 12∣ − 2 ?A. { Y ∣ Y \textgreater − 2 } \{y \mid Y \ \textgreater \ -2\} { Y ∣ Y \textgreater − 2 } B. { Y ∣ Y ≥ − 2 } \{y \mid Y \geq -2\} { Y ∣ Y ≥ − 2 } C. { Y ∣ Y \textgreater 12 } \{y \mid Y \ \textgreater \ 12\} { Y ∣ Y \textgreater 12 } D. { Y ∣ Y ≥ 12 } \{y \mid Y \geq 12\} { Y ∣ Y ≥ 12 }

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Introduction

When dealing with functions, understanding the range is crucial in determining the possible output values. The range of a function is the set of all possible output values it can produce for the given input values. In this article, we will explore the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2. We will analyze the function, identify its key characteristics, and determine the possible output values.

Understanding the Absolute Value Function

The function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 involves the absolute value of the expression x−12x - 12. The absolute value function ∣x∣|x| returns the distance of xx from zero on the number line. This means that for any real number xx, ∣x∣|x| is always non-negative.

Graphing the Function

To visualize the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2, we can graph it on a coordinate plane. The graph of the absolute value function ∣x−12∣|x - 12| is a V-shaped graph with its vertex at (12,0)(12, 0). The graph of g(x)=∣x−12∣−2g(x) = |x - 12| - 2 is a translation of the graph of ∣x−12∣|x - 12| down by 2 units.

Determining the Range

To determine the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2, we need to consider the possible output values. Since the graph of the function is a translation of the graph of ∣x−12∣|x - 12| down by 2 units, the minimum value of the function is −2-2. This occurs when x=12x = 12.

Analyzing the Graph

As we move away from x=12x = 12, the value of the function increases. Since the graph of the function is a V-shaped graph, the function increases without bound as xx approaches positive infinity or negative infinity. This means that the function can take on any value greater than or equal to −2-2.

Conclusion

In conclusion, the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 is the set of all possible output values it can produce for the given input values. Based on our analysis, we can determine that the range of the function is {y∣y≥−2}\{y \mid y \geq -2\}.

Final Answer

The final answer is {y∣y≥−2}\boxed{\{y \mid y \geq -2\}}.

Discussion

The range of a function is an essential concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we explored the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 and determined that the range is {y∣y≥−2}\{y \mid y \geq -2\}. We hope that this article has provided valuable insights into the concept of range and has helped readers understand the importance of analyzing functions to determine their range.

Related Topics

  • Range of a function
  • Absolute value function
  • Graphing functions
  • Translations of functions

References

  • [1] "Functions" by Khan Academy
  • [2] "Absolute Value Functions" by Math Open Reference
  • [3] "Graphing Functions" by Purplemath

Keywords

  • Range of a function
  • Absolute value function
  • Graphing functions
  • Translations of functions
  • Mathematics
  • Functions
  • Absolute value
  • Graphing
  • Translations

Introduction

In our previous article, we explored the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2. We determined that the range of the function is {y∣y≥−2}\{y \mid y \geq -2\}. In this article, we will answer some frequently asked questions about the range of the function and provide additional insights into the concept of range.

Q&A

Q1: What is the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2?

A1: The range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 is {y∣y≥−2}\{y \mid y \geq -2\}.

Q2: Why is the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 greater than or equal to −2-2?

A2: The range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 is greater than or equal to −2-2 because the graph of the function is a translation of the graph of ∣x−12∣|x - 12| down by 2 units. This means that the minimum value of the function is −2-2, which occurs when x=12x = 12.

Q3: Can the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 take on any value greater than −2-2?

A3: Yes, the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 can take on any value greater than −2-2. As xx approaches positive infinity or negative infinity, the value of the function increases without bound.

Q4: How does the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 compare to the range of the function f(x)=x2f(x) = x^2?

A4: The range of the function f(x)=x2f(x) = x^2 is {y∣y≥0}\{y \mid y \geq 0\}. This means that the range of the function f(x)=x2f(x) = x^2 is a subset of the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2. The range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 includes all non-negative values, while the range of the function f(x)=x2f(x) = x^2 only includes non-negative values that are greater than or equal to 0.

Q5: Can the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 be expressed as an interval?

A5: Yes, the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 can be expressed as an interval. The range of the function is {y∣y≥−2}\{y \mid y \geq -2\}, which can be expressed as the interval [−2,∞)[-2, \infty).

Conclusion

In conclusion, the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 is {y∣y≥−2}\{y \mid y \geq -2\}. We hope that this article has provided valuable insights into the concept of range and has helped readers understand the importance of analyzing functions to determine their range.

Final Answer

The final answer is {y∣y≥−2}\boxed{\{y \mid y \geq -2\}}.

Discussion

The range of a function is an essential concept in mathematics, and understanding it is crucial in solving problems involving functions. In this article, we explored the range of the function g(x)=∣x−12∣−2g(x) = |x - 12| - 2 and answered some frequently asked questions about the range of the function. We hope that this article has provided valuable insights into the concept of range and has helped readers understand the importance of analyzing functions to determine their range.

Related Topics

  • Range of a function
  • Absolute value function
  • Graphing functions
  • Translations of functions

References

  • [1] "Functions" by Khan Academy
  • [2] "Absolute Value Functions" by Math Open Reference
  • [3] "Graphing Functions" by Purplemath

Keywords

  • Range of a function
  • Absolute value function
  • Graphing functions
  • Translations of functions
  • Mathematics
  • Functions
  • Absolute value
  • Graphing
  • Translations