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 }

Understanding the Function

The given function is $g(x) = |x-12| - 2$. To find the range of this function, we need to understand its behavior. The absolute value function $|x-12|$ represents the distance of $x$ from 12 on the number line. When $x$ is less than 12, $|x-12|$ is the distance from $x$ to 12, and when $x$ is greater than 12, $|x-12|$ is the distance from $x$ to 12 as well.

Graphical Representation

Let's visualize the graph of the function $g(x) = |x-12| - 2$. We can see that the graph is a V-shaped graph with its vertex at (12, -2). The graph opens upwards, and its minimum value is -2.

Finding the Range

To find the range of the function, we need to find the set of all possible values that $g(x)$ can take. Since the graph of the function is a V-shaped graph, the range will be all the values that lie on or above the minimum value of the graph.

Analyzing the Options

Let's analyze the given options:

A. $\{y \mid y \ \textgreater \ -2\}$

B. $\{y \mid y \geq -2\}$

C. $\{y \mid y \ \textgreater \ 12\}$

D. $\{y \mid y \geq 12\}$

Option A

Option A states that the range is all the values greater than -2. However, this is not correct because the function can take the value -2 at $x=12$.

Option B

Option B states that the range is all the values greater than or equal to -2. This is the correct option because the function can take the value -2 at $x=12$, and all the values greater than -2 are possible.

Option C

Option C states that the range is all the values greater than 12. However, this is not correct because the function can take any value between -2 and infinity.

Option D

Option D states that the range is all the values greater than or equal to 12. However, this is not correct because the function can take any value between -2 and infinity.

Conclusion

In conclusion, the range of the function $g(x) = |x-12| - 2$ is all the values greater than or equal to -2. This is because the function can take the value -2 at $x=12$, and all the values greater than -2 are possible.

Final Answer

The final answer is B. $\{y \mid y \geq -2\}$.

Frequently Asked Questions

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

A1: The range of the function $g(x) = |x-12| - 2$ is all the values greater than or equal to -2. This is because the function can take the value -2 at $x=12$, and all the values greater than -2 are possible.

Q2: Why is the range of the function $g(x) = |x-12| - 2$ not all the values greater than -2?

A2: The range of the function $g(x) = |x-12| - 2$ is not all the values greater than -2 because the function can take the value -2 at $x=12$. This means that the value -2 is included in the range of the function.

Q3: What is the minimum value of the function $g(x) = |x-12| - 2$?

A3: The minimum value of the function $g(x) = |x-12| - 2$ is -2. This occurs when $x=12$.

Q4: Can the function $g(x) = |x-12| - 2$ take any value between -2 and infinity?

A4: Yes, the function $g(x) = |x-12| - 2$ can take any value between -2 and infinity. This is because the function is a V-shaped graph that opens upwards, and its minimum value is -2.

Q5: Is the range of the function $g(x) = |x-12| - 2$ all the values greater than or equal to 12?

A5: No, the range of the function $g(x) = |x-12| - 2$ is not all the values greater than or equal to 12. This is because the function can take any value between -2 and infinity, not just values greater than or equal to 12.

Q6: Can the function $g(x) = |x-12| - 2$ take any value less than -2?

A6: No, the function $g(x) = |x-12| - 2$ cannot take any value less than -2. This is because the function is a V-shaped graph that opens upwards, and its minimum value is -2.

Q7: What is the vertex of the function $g(x) = |x-12| - 2$?

A7: The vertex of the function $g(x) = |x-12| - 2$ is (12, -2). This is the point where the function changes from decreasing to increasing.

Q8: Is the function $g(x) = |x-12| - 2$ a linear function?

A8: No, the function $g(x) = |x-12| - 2$ is not a linear function. This is because it is a V-shaped graph that opens upwards, and its minimum value is -2.

Q9: Can the function $g(x) = |x-12| - 2$ be represented as a linear function?

A9: No, the function $g(x) = |x-12| - 2$ cannot be represented as a linear function. This is because it is a V-shaped graph that opens upwards, and its minimum value is -2.

Q10: What is the domain of the function $g(x) = |x-12| - 2$?

A10: The domain of the function $g(x) = |x-12| - 2$ is all real numbers. This is because the function is defined for all values of $x$.

Conclusion

In conclusion, the range of the function $g(x) = |x-12| - 2$ is all the values greater than or equal to -2. This is because the function can take the value -2 at $x=12$, and all the values greater than -2 are possible.