Select The Correct Answer.What Is The Range Of $g(x) = -\frac{1}{2}|x-6| + 1 ? ? ? A. $(-\infty, 1] B. {1, \infty }$ C. {6, \infty }$ D. ( − ∞ , ∞ (-\infty, \infty ( − ∞ , ∞ ]

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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. In this article, we will focus on finding the range of the function $g(x) = -\frac{1}{2}|x-6| + 1$. This function involves absolute value, which can make it challenging to determine its range. We will break down the process into manageable steps and provide a clear explanation of each step.

What is the Absolute Value Function?

Before we dive into the function $g(x) = -\frac{1}{2}|x-6| + 1$, let's briefly discuss the absolute value function. The absolute value function, denoted by $|x|$, is defined as:

∣x∣={x,if x≥0−x,if x<0|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

In other words, the absolute value of a number is its distance from zero on the number line. For example, $|5| = 5$ and $|-3| = 3$.

Analyzing the Function $g(x) = -\frac{1}{2}|x-6| + 1$

Now that we have a basic understanding of the absolute value function, let's analyze the function $g(x) = -\frac{1}{2}|x-6| + 1$. This function involves two main components:

  1. The absolute value term $|x-6|$
  2. The constant term $1$

Step 1: Understanding the Absolute Value Term

The absolute value term $|x-6|$ represents the distance between $x$ and $6$ on the number line. This term is always non-negative, as it represents a distance.

Step 2: Understanding the Constant Term

The constant term $1$ is simply added to the absolute value term. This means that the output of the function will always be at least $1$.

Step 3: Combining the Terms

Now that we have analyzed the individual terms, let's combine them to understand the behavior of the function. The function $g(x) = -\frac{1}{2}|x-6| + 1$ can be rewritten as:

g(x)={−12(x−6)+1,if x≥6−12(−x+6)+1,if x<6g(x) = \begin{cases} -\frac{1}{2}(x-6) + 1, & \text{if } x \geq 6 \\ -\frac{1}{2}(-x+6) + 1, & \text{if } x < 6 \end{cases}

Simplifying the Function

Let's simplify the function by evaluating the two cases:

Case 1: $x \geq 6$

g(x)=−12(x−6)+1g(x) = -\frac{1}{2}(x-6) + 1

g(x)=−12x+3+1g(x) = -\frac{1}{2}x + 3 + 1

g(x)=−12x+4g(x) = -\frac{1}{2}x + 4

Case 2: $x < 6$

g(x)=−12(−x+6)+1g(x) = -\frac{1}{2}(-x+6) + 1

g(x)=12x−3+1g(x) = \frac{1}{2}x - 3 + 1

g(x)=12x−2g(x) = \frac{1}{2}x - 2

Understanding the Behavior of the Function

Now that we have simplified the function, let's understand its behavior. The function $g(x) = -\frac{1}{2}x + 4$ for $x \geq 6$ is a decreasing linear function, while the function $g(x) = \frac{1}{2}x - 2$ for $x < 6$ is an increasing linear function.

Determining the Range of the Function

To determine the range of the function, we need to find the minimum and maximum values it can produce. Let's analyze the two cases:

Case 1: $x \geq 6$

The function $g(x) = -\frac{1}{2}x + 4$ is a decreasing linear function, which means that its minimum value occurs at the maximum value of $x$. Since $x \geq 6$, the maximum value of $x$ is $\infty$. Therefore, the minimum value of the function is:

lim⁡x→∞−12x+4=−∞\lim_{x \to \infty} -\frac{1}{2}x + 4 = -\infty

However, since the function is decreasing, it will never actually reach $-\infty$. Instead, it will approach $-\infty$ as $x$ approaches $\infty$. Therefore, the range of the function for $x \geq 6$ is $(-\infty, 4]$.

Case 2: $x < 6$

The function $g(x) = \frac{1}{2}x - 2$ is an increasing linear function, which means that its minimum value occurs at the minimum value of $x$. Since $x < 6$, the minimum value of $x$ is $-\infty$. Therefore, the minimum value of the function is:

lim⁡x→−∞12x−2=−∞\lim_{x \to -\infty} \frac{1}{2}x - 2 = -\infty

However, since the function is increasing, it will never actually reach $-\infty$. Instead, it will approach $-\infty$ as $x$ approaches $-\infty$. Therefore, the range of the function for $x < 6$ is $(-\infty, 4]$.

Combining the Ranges

Since the function is defined differently for $x \geq 6$ and $x < 6$, we need to combine the ranges for each case. The range of the function for $x \geq 6$ is $(-\infty, 4]$, while the range of the function for $x < 6$ is also $(-\infty, 4]$.

Therefore, the range of the function $g(x) = -\frac{1}{2}|x-6| + 1$ is $(-\infty, 4]$.

Conclusion

In this article, we analyzed the function $g(x) = -\frac{1}{2}|x-6| + 1$ and determined its range. We broke down the process into manageable steps and provided a clear explanation of each step. We simplified the function, understood its behavior, and determined the range of the function. The range of the function is $(-\infty, 4]$.

Answer

Introduction

In our previous article, we analyzed the function $g(x) = -\frac{1}{2}|x-6| + 1$ and determined its range. We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will answer some frequently asked questions related to the range of a function.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values it can produce for the given input values.

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

A: To determine the range of a function, you need to analyze the function and identify its minimum and maximum values. You can do this by graphing the function, using algebraic methods, or using calculus.

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

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have a range of $(-\infty, \infty)$?

A: Yes, a function can have a range of $(-\infty, \infty)$. This occurs when the function is defined for all real numbers and has no maximum or minimum value.

Q: How do I determine if a function has a range of $(-\infty, \infty)$?

A: To determine if a function has a range of $(-\infty, \infty)$, you need to analyze the function and identify its behavior. If the function is defined for all real numbers and has no maximum or minimum value, then it has a range of $(-\infty, \infty)$.

Q: Can a function have a range of $(-\infty, a]$?

A: Yes, a function can have a range of $(-\infty, a]$, where $a$ is a real number. This occurs when the function has a maximum value of $a$ and no minimum value.

Q: How do I determine if a function has a range of $(-\infty, a]$?

A: To determine if a function has a range of $(-\infty, a]$, you need to analyze the function and identify its maximum value. If the function has a maximum value of $a$ and no minimum value, then it has a range of $(-\infty, a]$.

Q: Can a function have a range of $[a, b]$?

A: Yes, a function can have a range of $[a, b]$, where $a$ and $b$ are real numbers. This occurs when the function has a minimum value of $a$ and a maximum value of $b$.

Q: How do I determine if a function has a range of $[a, b]$?

A: To determine if a function has a range of $[a, b]$, you need to analyze the function and identify its minimum and maximum values. If the function has a minimum value of $a$ and a maximum value of $b$, then it has a range of $[a, b]$.

Conclusion

In this article, we answered some frequently asked questions related to the range of a function. We provided a clear explanation of each question and provided examples to illustrate the concepts. We hope that this article has been helpful in understanding the range of a function.

Additional Resources

Final Answer

The final answer is:

  • Q: What is the range of a function? A: The range of a function is the set of all possible output values it can produce for the given input values.
  • Q: How do I determine the range of a function? A: To determine the range of a function, you need to analyze the function and identify its minimum and maximum values.
  • Q: Can a function have a range of $(-\infty, \infty)$? A: Yes, a function can have a range of $(-\infty, \infty)$.
  • Q: How do I determine if a function has a range of $(-\infty, \infty)$? A: To determine if a function has a range of $(-\infty, \infty)$, you need to analyze the function and identify its behavior.
  • Q: Can a function have a range of $(-\infty, a]$? A: Yes, a function can have a range of $(-\infty, a]$.
  • Q: How do I determine if a function has a range of $(-\infty, a]$? A: To determine if a function has a range of $(-\infty, a]$, you need to analyze the function and identify its maximum value.
  • Q: Can a function have a range of $[a, b]$? A: Yes, a function can have a range of $[a, b]$.
  • Q: How do I determine if a function has a range of $[a, b]$? A: To determine if a function has a range of $[a, b]$, you need to analyze the function and identify its minimum and maximum values.