The Functions F F F And G G G Are Defined As Follows: F ( X ) = X − 1 X 2 + 2 X + 1 G ( X ) = X + 6 X 2 − 2 X − 48 \begin{array}{l} f(x) = \frac{x-1}{x^2 + 2x + 1} \\ g(x) = \frac{x+6}{x^2 - 2x - 48} \end{array} F ( X ) = X 2 + 2 X + 1 X − 1 ​ G ( X ) = X 2 − 2 X − 48 X + 6 ​ ​ For Each Function, Find The Domain.Write Each Answer As An Interval Or Union

Introduction

In this article, we will explore the functions $f$ and $g$, which are defined as follows:

$\begin{array}{l} f(x) = \frac{x-1}{x^2 + 2x + 1} \\ g(x) = \frac{x+6}{x^2 - 2x - 48} \end{array}$

Our goal is to find the domain of each function, which is the set of all possible input values for which the function is defined.

Domain of a Function

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 $x$ for which the function is a real number.

To find the domain of a function, we need to consider the following:

• Denominator cannot be zero: The denominator of a fraction cannot be zero, as division by zero is undefined.
• No square roots of negative numbers: The expression inside a square root cannot be negative, as square roots of negative numbers are not real numbers.

Finding the Domain of f(x)

Let's start by finding the domain of $f(x)$.

Denominator of f(x)

The denominator of $f(x)$ is $x^2 + 2x + 1$. To find the values of $x$ for which the denominator is not zero, we need to solve the equation:

$x^2 + 2x + 1 = 0$

We can factor the left-hand side of the equation as:

$(x + 1)^2 = 0$

This implies that $x + 1 = 0$, which gives us:

$x = -1$

Therefore, the denominator of $f(x)$ is zero when $x = -1$.

Domain of f(x)

Since the denominator of $f(x)$ is zero when $x = -1$, we need to exclude this value from the domain of $f(x)$.

The domain of $f(x)$ is therefore:

$(-\infty, -1) \cup (-1, \infty)$

This means that $f(x)$ is defined for all real numbers except $x = -1$.

Finding the Domain of g(x)

Now, let's find the domain of $g(x)$.

Denominator of g(x)

The denominator of $g(x)$ is $x^2 - 2x - 48$. To find the values of $x$ for which the denominator is not zero, we need to solve the equation:

$x^2 - 2x - 48 = 0$

We can factor the left-hand side of the equation as:

$(x - 8)(x + 6) = 0$

This implies that either $x - 8 = 0$ or $x + 6 = 0$, which gives us:

$x = 8$ or $x = -6$

Therefore, the denominator of $g(x)$ is zero when $x = 8$ or $x = -6$.

Domain of g(x)

Since the denominator of $g(x)$ is zero when $x = 8$ or $x = -6$, we need to exclude these values from the domain of $g(x)$.

The domain of $g(x)$ is therefore:

$(-\infty, -6) \cup (-6, 8) \cup (8, \infty)$

This means that $g(x)$ is defined for all real numbers except $x = 8$ and $x = -6$.

Conclusion

In this article, we found the domain of the functions $f$ and $g$. The domain of $f(x)$ is $(-\infty, -1) \cup (-1, \infty)$, and the domain of $g(x)$ is $(-\infty, -6) \cup (-6, 8) \cup (8, \infty)$.

We hope this article has been helpful in understanding the concept of domain and how to find it for a given function.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak

Discussion

What are some other ways to find the domain of a function? How do you think the concept of domain is used in real-world applications?

Related Topics

• Finding the range of a function
• Graphing functions
• Solving equations and inequalities

Further Reading

• "Algebra and Trigonometry" by Michael Sullivan
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Q&A: The Functions f and g =============================

Introduction

In our previous article, we explored the functions $f$ and $g$, which are defined as follows:

$\begin{array}{l} f(x) = \frac{x-1}{x^2 + 2x + 1} \\ g(x) = \frac{x+6}{x^2 - 2x - 48} \end{array}$

We found the domain of each function, which is the set of all possible input values for which the function is defined.

In this article, we will answer some frequently asked questions about the functions $f$ and $g$.

Q: What is the domain of f(x)?

A: The domain of $f(x)$ is $(-\infty, -1) \cup (-1, \infty)$.

Q: What is the domain of g(x)?

A: The domain of $g(x)$ is $(-\infty, -6) \cup (-6, 8) \cup (8, \infty)$.

Q: Why is the denominator of f(x) zero when x = -1?

A: The denominator of $f(x)$ is zero when $x = -1$ because $x^2 + 2x + 1 = (-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$.

Q: Why is the denominator of g(x) zero when x = 8 or x = -6?

A: The denominator of $g(x)$ is zero when $x = 8$ or $x = -6$ because $x^2 - 2x - 48 = (8)^2 - 2(8) - 48 = 64 - 16 - 48 = 0$ and $(x + 6)^2 = (-6)^2 + 2(-6) + 36 = 36 - 12 + 36 = 60$.

Q: How do you find the domain of a function?

A: To find the domain of a function, you need to consider the following:

• Denominator cannot be zero: The denominator of a fraction cannot be zero, as division by zero is undefined.
• No square roots of negative numbers: The expression inside a square root cannot be negative, as square roots of negative numbers are not real numbers.

Q: What are some other ways to find the domain of a function?

A: Some other ways to find the domain of a function include:

• Graphing the function: Graphing the function can help you visualize the domain.
• Using algebraic methods: Using algebraic methods, such as factoring and solving equations, can help you find the domain.
• Using numerical methods: Using numerical methods, such as numerical integration and approximation, can help you find the domain.

Q: How do you think the concept of domain is used in real-world applications?

A: The concept of domain is used in many real-world applications, such as:

• Computer programming: In computer programming, the domain of a function is used to determine the input values that are valid for the function.
• Engineering: In engineering, the domain of a function is used to determine the input values that are valid for the function, and to ensure that the function is defined for all possible input values.
• Science: In science, the domain of a function is used to determine the input values that are valid for the function, and to ensure that the function is defined for all possible input values.

Conclusion

In this article, we answered some frequently asked questions about the functions $f$ and $g$. We hope this article has been helpful in understanding the concept of domain and how to find it for a given function.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak

Discussion

What are some other ways to find the domain of a function? How do you think the concept of domain is used in real-world applications?

Related Topics

• Finding the range of a function
• Graphing functions
• Solving equations and inequalities

Further Reading

• "Algebra and Trigonometry" by Michael Sullivan
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton