What Are The Domain And Range Of The Function F ( X ) = X 2 − 3 X − 28 X + 4 F(x)=\frac{x^2-3x-28}{x+4} F ( X ) = X + 4 X 2 − 3 X − 28 ​ ?A. D : { X ∈ R ∣ X ≠ 4 } , R : { Y ∈ R ∣ Y ≠ 11 } D:\{x \in \mathbb{R} \mid X \neq 4\}, R:\{y \in \mathbb{R} \mid Y \neq 11\} D : { X ∈ R ∣ X  = 4 } , R : { Y ∈ R ∣ Y  = 11 } B. $D:{x \in \mathbb R} \mid X \neq 7}, R {y \in \mathbb{R \mid Y \neq

Introduction

When dealing with functions, it's essential to understand the concept of domain and range. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this article, we will explore the domain and range of the function $f(x)=\frac{x^2-3x-28}{x+4}$.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is $x^2-3x-28$, and the denominator is $x+4$. To find the domain and range of this function, we need to consider the values of $x$ for which the function is defined.

Finding the Domain

The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero. In this case, the denominator is $x+4$, so we need to find the value of $x$ that makes $x+4=0$. Solving for $x$, we get:

$$x+4=0 \Rightarrow x=-4$$

Therefore, the domain of the function $f(x)$ is all real numbers except $x=-4$. We can write this as:

$$D:\{x \in \mathbb{R} \mid x \neq -4\}$$

Finding the Range

To find the range of the function, we need to consider the possible output values of the function. Since the function is a rational function, we can use the concept of limits to find the range.

Let's consider the limit of the function as $x$ approaches $-4$:

$$\lim_{x \to -4} f(x) = \lim_{x \to -4} \frac{x^2-3x-28}{x+4}$$

Using the fact that the numerator and denominator both approach zero as $x$ approaches $-4$, we can use L'Hopital's rule to find the limit:

$$\lim_{x \to -4} f(x) = \lim_{x \to -4} \frac{2x-3}{1} = \frac{2(-4)-3}{1} = -11$$

Therefore, the range of the function $f(x)$ is all real numbers except $y=-11$. We can write this as:

$$R:\{y \in \mathbb{R} \mid y \neq -11\}$$

Conclusion

In conclusion, the domain of the function $f(x)=\frac{x^2-3x-28}{x+4}$ is all real numbers except $x=-4$, and the range is all real numbers except $y=-11$. Therefore, the correct answer is:

$$D:\{x \in \mathbb{R} \mid x \neq -4\}, R:\{y \in \mathbb{R} \mid y \neq -11\}$$

Final Answer

The final answer is $D:\{x \in \mathbb{R} \mid x \neq -4\}, R:\{y \in \mathbb{R} \mid y \neq -11\}$.

Introduction

In our previous article, we explored the domain and range of the function $f(x)=\frac{x^2-3x-28}{x+4}$. In this article, we will answer some frequently asked questions related to the domain and range of this function.

Q1: What is the domain of the function $f(x)=\frac{x^2-3x-28}{x+4}$?

A1: The domain of the function $f(x)=\frac{x^2-3x-28}{x+4}$ is all real numbers except $x=-4$. This is because the denominator of the function is $x+4$, and we cannot divide by zero.

Q2: Why is the domain of the function $f(x)=\frac{x^2-3x-28}{x+4}$ not all real numbers?

A2: The domain of the function $f(x)=\frac{x^2-3x-28}{x+4}$ is not all real numbers because the denominator of the function is $x+4$, and we cannot divide by zero. When $x=-4$, the denominator becomes zero, and the function is undefined.

Q3: What is the range of the function $f(x)=\frac{x^2-3x-28}{x+4}$?

A3: The range of the function $f(x)=\frac{x^2-3x-28}{x+4}$ is all real numbers except $y=-11$. This is because the limit of the function as $x$ approaches $-4$ is $-11$, and the function is continuous for all other values of $x$.

Q4: Why is the range of the function $f(x)=\frac{x^2-3x-28}{x+4}$ not all real numbers?

A4: The range of the function $f(x)=\frac{x^2-3x-28}{x+4}$ is not all real numbers because the function is undefined at $x=-4$, and the limit of the function as $x$ approaches $-4$ is $-11$. This means that the function does not take on the value $-11$ for any value of $x$.

Q5: How do I find the domain and range of a rational function?

A5: To find the domain and range of a rational function, you need to consider the values of $x$ for which the function is defined. This means that you need to find the values of $x$ that make the denominator equal to zero, and then exclude those values from the domain. You also need to consider the limit of the function as $x$ approaches those values, and exclude any values that the function approaches but does not take on.

Q6: What is the significance of the domain and range of a function?

A6: The domain and range of a function are important because they tell us the possible input and output values of the function. This is useful for a variety of applications, including graphing the function, solving equations, and analyzing the behavior of the function.

Q7: How do I graph a rational function?

A7: To graph a rational function, you need to consider the domain and range of the function. You can use the domain and range to determine the x-intercepts and y-intercepts of the function, and then use those points to graph the function.

Q8: What is the difference between the domain and range of a function?

A8: The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. This means that the domain tells us the possible values of $x$ that we can plug into the function, while the range tells us the possible values of $y$ that the function can take on.

Q9: How do I find the domain and range of a function with a square root in the denominator?

A9: To find the domain and range of a function with a square root in the denominator, you need to consider the values of $x$ for which the denominator is non-negative. This means that you need to find the values of $x$ that make the square root equal to zero, and then exclude those values from the domain.

Q10: What is the significance of the domain and range of a function in real-world applications?

A10: The domain and range of a function are important in real-world applications because they tell us the possible input and output values of the function. This is useful for a variety of applications, including modeling population growth, analyzing financial data, and predicting weather patterns.

Conclusion

In conclusion, the domain and range of a function are important concepts that tell us the possible input and output values of the function. By understanding the domain and range of a function, we can graph the function, solve equations, and analyze the behavior of the function.