For The Function $y=\frac{1}{|5-x|}$, What Is The Domain?A. $(0, \infty$\] And $x \neq 5$ B. $(-\infty, \infty$\] And $x \neq 5$ C. $(-\infty, \infty$\] And $x \neq 0$ D. \[- \infty,

Introduction

When dealing with rational functions, it's essential to understand the concept of domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this article, we will explore the domain of the function $y=\frac{1}{|5-x|}$ and examine the different options provided.

Understanding Absolute Value

Before diving into the domain of the function, let's briefly discuss absolute value. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of -3 is 3, and the absolute value of 4 is 4. The absolute value function is denoted by |x| and is defined as:

|x| = x if x ≥ 0 |x| = -x if x < 0

The Function $y=\frac{1}{|5-x|}$

The given function is $y=\frac{1}{|5-x|}$. To find the domain of this function, we need to consider the values of x that make the denominator zero. Since the denominator is an absolute value expression, we need to consider two cases:

1. When $5-x \geq 0$, the absolute value expression simplifies to $|5-x| = 5-x$.
2. When $5-x < 0$, the absolute value expression simplifies to $|5-x| = -(5-x) = x-5$.

Case 1: $5-x \geq 0$

In this case, the absolute value expression simplifies to $|5-x| = 5-x$. The denominator of the function is $|5-x|$, which is equal to $5-x$. To find the values of x that make the denominator zero, we set $5-x = 0$ and solve for x:

$$5-x = 0$$

$$x = 5$$

However, this value of x is not in the domain of the function because it makes the denominator zero.

Case 2: $5-x < 0$

In this case, the absolute value expression simplifies to $|5-x| = x-5$. The denominator of the function is $|5-x|$, which is equal to $x-5$. To find the values of x that make the denominator zero, we set $x-5 = 0$ and solve for x:

$$x-5 = 0$$

$$x = 5$$

However, this value of x is not in the domain of the function because it makes the denominator zero.

Conclusion

Based on the analysis of the two cases, we can conclude that the function $y=\frac{1}{|5-x|}$ is defined for all real numbers except x = 5. Therefore, the domain of the function is all real numbers except x = 5.

Answer

The correct answer is:

• B. $(-\infty, \infty)$ and $x \neq 5$

This answer choice correctly represents the domain of the function $y=\frac{1}{|5-x|}$.

Final Thoughts

Introduction

In our previous article, we explored the concept of domain and applied it to the function $y=\frac{1}{|5-x|}$. We concluded that the domain of the function is all real numbers except x = 5. In this article, we will address some common questions and concerns related to the domain of a rational function.

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all real numbers for which the function is not undefined.

Q: How do I determine the domain of a rational function?

A: To determine the domain of a rational function, you need to consider the values of x that make the denominator zero. You can do this by setting the denominator equal to zero and solving for x. Any values of x that make the denominator zero must be excluded from the domain.

Q: What happens if the denominator is a constant?

A: If the denominator is a constant, then the function is defined for all real numbers except the value that makes the denominator zero. For example, if the function is $y=\frac{1}{2}$, then the domain is all real numbers except x = 2.

Q: Can the domain of a rational function be a single value?

A: Yes, the domain of a rational function can be a single value. For example, if the function is $y=\frac{1}{x-5}$, then the domain is x = 5.

Q: How do I represent the domain of a rational function?

A: The domain of a rational function can be represented in several ways, including:

• Using interval notation: For example, the domain of the function $y=\frac{1}{x-5}$ can be represented as (-∞, 5) ∪ (5, ∞).
• Using set notation: For example, the domain of the function $y=\frac{1}{x-5}$ can be represented as {x | x ≠ 5}.
• Using a graph: For example, the domain of the function $y=\frac{1}{x-5}$ can be represented as a graph with a vertical asymptote at x = 5.

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

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) for which the function is defined.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, if the function is $y=x$, then the domain and range are both all real numbers.

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values (x-values) for which the function is defined. By considering the values of x that make the denominator zero, we can determine the domain of a rational function. We hope this article has helped to clarify any questions or concerns you may have had about the domain of a rational function.

Frequently Asked Questions

• Q: What is the domain of a rational function? A: The domain of a rational function is the set of all possible input values (x-values) for which the function is defined.
• Q: How do I determine the domain of a rational function? A: To determine the domain of a rational function, you need to consider the values of x that make the denominator zero.
• Q: What happens if the denominator is a constant? A: If the denominator is a constant, then the function is defined for all real numbers except the value that makes the denominator zero.

Additional Resources

• Khan Academy: Domain and Range A video tutorial on the domain and range of a function.
• Mathway: Domain and Range A online calculator that can help you determine the domain and range of a function.
• Wolfram Alpha: Domain and Range A online calculator that can help you determine the domain and range of a function.