Find The Domain Of H ( X H(x H ( X ] And Write Your Answer In Set Notation And Interval Notation. Round Answers To 3 Decimal Places As Needed. H ( X ) = ( 2 X − 5 ) ( 8 X + 5 ) 3 ( X + 1 ) ( − 5 X − 8 ) 2 ( 2 X − 5 ) ( X − 9 ) 2 H(x) = \frac{(2x-5)(8x+5)^3(x+1)}{(-5x-8)^2(2x-5)(x-9)^2} H ( X ) = ( − 5 X − 8 ) 2 ( 2 X − 5 ) ( X − 9 ) 2 ( 2 X − 5 ) ( 8 X + 5 ) 3 ( X + 1 ) ​ Set Notation: { X ∣ X \{x \mid X { X ∣ X Is A

$$

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of $x$ that can be plugged into the function without causing any problems. In this article, we will find the domain of the given function $h(x)$ and express it in both set notation and interval notation.

The Function $h(x)$

The given function is:

$$h(x) = \frac{(2x-5)(8x+5)^3(x+1)}{(-5x-8)^2(2x-5)(x-9)^2}$$

Finding the Domain

To find the domain of $h(x)$, we need to identify the values of $x$ that make the function undefined. In other words, we need to find the values of $x$ that make the denominator of the function equal to zero.

Setting the Denominator Equal to Zero

Let's set the denominator equal to zero and solve for $x$:

$$(-5x-8)^2(2x-5)(x-9)^2 = 0$$

Solving the Equation

To solve the equation, we need to find the values of $x$ that make each factor equal to zero. Let's start with the first factor:

$$(-5x-8)^2 = 0$$

This implies that:

$$-5x-8 = 0$$

Solving for $x$, we get:

$$x = -\frac{8}{5}$$

Second Factor

Now, let's consider the second factor:

$$(2x-5) = 0$$

Solving for $x$, we get:

$$x = \frac{5}{2}$$

Third Factor

Finally, let's consider the third factor:

$$(x-9)^2 = 0$$

This implies that:

$$x-9 = 0$$

Solving for $x$, we get:

$$x = 9$$

Domain in Set Notation

The domain of $h(x)$ is the set of all values of $x$ that are not equal to $-\frac{8}{5}$, $\frac{5}{2}$, or $9$. In set notation, this can be written as:

$$\{x \mid x \neq -\frac{8}{5}, \frac{5}{2}, 9\}$$

Domain in Interval Notation

To express the domain in interval notation, we need to consider the intervals between the values of $x$ that make the function undefined. The domain of $h(x)$ is the union of the following intervals:

$$(-\infty, -\frac{8}{5}) \cup (-\frac{8}{5}, \frac{5}{2}) \cup (\frac{5}{2}, 9) \cup (9, \infty)$$

Conclusion

In this article, we found the domain of the given function $h(x)$ and expressed it in both set notation and interval notation. The domain of $h(x)$ is the set of all values of $x$ that are not equal to $-\frac{8}{5}$, $\frac{5}{2}$, or $9$. We also expressed the domain in interval notation as the union of the intervals $(-\infty, -\frac{8}{5}) \cup (-\frac{8}{5}, \frac{5}{2}) \cup (\frac{5}{2}, 9) \cup (9, \infty)$.

Final Answer

The final answer is $\boxed{(-\infty, -\frac{8}{5}) \cup (-\frac{8}{5}, \frac{5}{2}) \cup (\frac{5}{2}, 9) \cup (9, \infty)}$.

$$

Introduction

In our previous article, we found the domain of the given function $h(x)$ and expressed it in both set notation and interval notation. In this article, we will answer some frequently asked questions about the domain of $h(x)$.

Q: What is the domain of $h(x)$?

A: The domain of $h(x)$ is the set of all values of $x$ that are not equal to $-\frac{8}{5}$, $\frac{5}{2}$, or $9$. In set notation, this can be written as:

$$\{x \mid x \neq -\frac{8}{5}, \frac{5}{2}, 9\}$$

Q: How do I express the domain in interval notation?

A: To express the domain in interval notation, we need to consider the intervals between the values of $x$ that make the function undefined. The domain of $h(x)$ is the union of the following intervals:

$$(-\infty, -\frac{8}{5}) \cup (-\frac{8}{5}, \frac{5}{2}) \cup (\frac{5}{2}, 9) \cup (9, \infty)$$

Q: Why are the values $-\frac{8}{5}$, $\frac{5}{2}$, and $9$ excluded from the domain?

A: The values $-\frac{8}{5}$, $\frac{5}{2}$, and $9$ are excluded from the domain because they make the denominator of the function equal to zero. When the denominator is equal to zero, the function is undefined.

Q: Can I simplify the domain of $h(x)$?

A: Yes, you can simplify the domain of $h(x)$ by combining the intervals. However, the simplified domain will still be the same as the original domain.

Q: How do I find the domain of a function with multiple factors in the denominator?

A: To find the domain of a function with multiple factors in the denominator, you need to set each factor equal to zero and solve for $x$. The values of $x$ that make each factor equal to zero are the values that are excluded from the domain.

Q: Can I use a calculator to find the domain of $h(x)$?

A: Yes, you can use a calculator to find the domain of $h(x)$. However, you need to be careful when using a calculator to ensure that you get the correct answer.

Q: What is the significance of the domain of a function?

A: The domain of a function is significant because it tells us the set of all possible input values for which the function is defined. In other words, it tells us the set of all possible values of $x$ that can be plugged into the function without causing any problems.

Q: Can I find the domain of a function with a rational expression?

A: Yes, you can find the domain of a function with a rational expression by setting the denominator equal to zero and solving for $x$. The values of $x$ that make the denominator equal to zero are the values that are excluded from the domain.

Q: How do I find the domain of a function with a radical expression?

A: To find the domain of a function with a radical expression, you need to consider the values of $x$ that make the expression inside the radical equal to zero or negative. The values of $x$ that make the expression inside the radical equal to zero or negative are the values that are excluded from the domain.

Q: Can I find the domain of a function with a trigonometric expression?

A: Yes, you can find the domain of a function with a trigonometric expression by considering the values of $x$ that make the expression inside the trigonometric function equal to zero or undefined. The values of $x$ that make the expression inside the trigonometric function equal to zero or undefined are the values that are excluded from the domain.

Q: How do I find the domain of a function with a piecewise expression?

A: To find the domain of a function with a piecewise expression, you need to consider the values of $x$ that make each piece of the expression undefined. The values of $x$ that make each piece of the expression undefined are the values that are excluded from the domain.

Q: Can I find the domain of a function with a parametric expression?

A: Yes, you can find the domain of a function with a parametric expression by considering the values of the parameter that make the expression undefined. The values of the parameter that make the expression undefined are the values that are excluded from the domain.

Q: How do I find the domain of a function with a polar expression?

A: To find the domain of a function with a polar expression, you need to consider the values of the polar angle that make the expression undefined. The values of the polar angle that make the expression undefined are the values that are excluded from the domain.

Conclusion

In this article, we answered some frequently asked questions about the domain of $h(x)$. We hope that this article has been helpful in clarifying any doubts you may have had about the domain of $h(x)$. If you have any further questions, please don't hesitate to ask.