The Functions F F F And G G G Are Defined As F ( X ) = 3 X + 10 F(x) = 3x + 10 F ( X ) = 3 X + 10 And G ( X ) = 9 − 10 X G(x) = 9 - 10x G ( X ) = 9 − 10 X .a) Find The Domain Of F F F , G G G , F + G F+g F + G , F − G F-g F − G , F G Fg F G , F F Ff Ff , F G \frac{f}{g} G F ​ ,

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Introduction

In this article, we will explore the domain of two functions, $f(x) = 3x + 10$ and $g(x) = 9 - 10x$, and their various compositions. The domain of a function is the set of all possible input values for which the function is defined. Understanding the domain of a function is crucial in mathematics, as it helps us determine the validity of a function's output.

Domain of $f(x) = 3x + 10$

The function $f(x) = 3x + 10$ is a linear function, which means it is defined for all real numbers. In other words, the domain of $f(x)$ is the set of all real numbers, denoted as $\mathbb{R}$.

Domain of $g(x) = 9 - 10x$

The function $g(x) = 9 - 10x$ is also a linear function, but it has a restriction on its domain. Since the function involves a subtraction operation, we must ensure that the expression $9 - 10x$ is defined. This means that $x$ cannot be equal to $\frac{9}{10}$, as it would result in a division by zero. Therefore, the domain of $g(x)$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Domain of $f+g$

The function $f+g$ is the sum of $f(x)$ and $g(x)$. To find the domain of $f+g$, we need to consider the restrictions on both $f(x)$ and $g(x)$. Since $f(x)$ is defined for all real numbers, and $g(x)$ is defined for all real numbers except $\frac{9}{10}$, the domain of $f+g$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Domain of $f-g$

The function $f-g$ is the difference of $f(x)$ and $g(x)$. To find the domain of $f-g$, we need to consider the restrictions on both $f(x)$ and $g(x)$. Since $f(x)$ is defined for all real numbers, and $g(x)$ is defined for all real numbers except $\frac{9}{10}$, the domain of $f-g$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Domain of $fg$

The function $fg$ is the product of $f(x)$ and $g(x)$. To find the domain of $fg$, we need to consider the restrictions on both $f(x)$ and $g(x)$. Since $f(x)$ is defined for all real numbers, and $g(x)$ is defined for all real numbers except $\frac{9}{10}$, the domain of $fg$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Domain of $ff$

The function $ff$ is the composition of $f(x)$ with itself. To find the domain of $ff$, we need to consider the restrictions on $f(x)$. Since $f(x)$ is defined for all real numbers, the domain of $ff$ is the set of all real numbers, denoted as $\mathbb{R}$.

Domain of $\frac{f}{g}$

The function $\frac{f}{g}$ is the quotient of $f(x)$ and $g(x)$. To find the domain of $\frac{f}{g}$, we need to consider the restrictions on both $f(x)$ and $g(x)$. Since $f(x)$ is defined for all real numbers, and $g(x)$ is defined for all real numbers except $\frac{9}{10}$, the domain of $\frac{f}{g}$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Conclusion

In conclusion, the domain of $f(x) = 3x + 10$ is the set of all real numbers, denoted as $\mathbb{R}$. The domain of $g(x) = 9 - 10x$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$. The domains of $f+g$, $f-g$, $fg$, and $\frac{f}{g}$ are also the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$. The domain of $ff$ is the set of all real numbers, denoted as $\mathbb{R}$.

References

• [1] "Functions and Their Compositions" by [Author's Name]
• [2] "Domain and Range of Functions" by [Author's Name]

Further Reading

• [1] "Linear Functions" by [Author's Name]
• [2] "Quadratic Functions" by [Author's Name]

Glossary

• Domain: The set of all possible input values for which a function is defined.
• Range: The set of all possible output values for which a function is defined.
• Composition: The process of combining two or more functions to create a new function.
• Linear Function: A function that can be written in the form $f(x) = mx + b$, where $m$ and $b$ are constants.
• Quadratic Function: A function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
  Q&A: Functions $f$ and $g$ =============================

Q: What is the domain of $f(x) = 3x + 10$?

A: The domain of $f(x) = 3x + 10$ is the set of all real numbers, denoted as $\mathbb{R}$.

Q: What is the domain of $g(x) = 9 - 10x$?

A: The domain of $g(x) = 9 - 10x$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Q: What is the domain of $f+g$?

A: The domain of $f+g$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Q: What is the domain of $f-g$?

A: The domain of $f-g$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Q: What is the domain of $fg$?

A: The domain of $fg$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Q: What is the domain of $ff$?

A: The domain of $ff$ is the set of all real numbers, denoted as $\mathbb{R}$.

Q: What is the domain of $\frac{f}{g}$?

A: The domain of $\frac{f}{g}$ is the set of all real numbers except $\frac{9}{10}$, denoted as $\mathbb{R} \setminus \{\frac{9}{10}\}$.

Q: Why is the domain of $g(x) = 9 - 10x$ restricted?

A: The domain of $g(x) = 9 - 10x$ is restricted because the expression $9 - 10x$ is undefined when $x = \frac{9}{10}$, as it would result in a division by zero.

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

A: The domain of a function is significant because it determines the set of all possible input values for which the function is defined. Understanding the domain of a function is crucial in mathematics, as it helps us determine the validity of a function's output.

Q: Can the domain of a function be changed?

A: No, the domain of a function cannot be changed. The domain of a function is a fixed set of values that defines the function's behavior.

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

A: To find the domain of a function, you need to consider the restrictions on the function's input values. This may involve identifying any values that would result in a division by zero, a square root of a negative number, or any other undefined mathematical operation.

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 for which the function is defined, while the range of a function is the set of all possible output values for which the function is defined.

Q: Can the range of a function be changed?

A: No, the range of a function cannot be changed. The range of a function is a fixed set of values that defines the function's output behavior.

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

A: To find the range of a function, you need to consider the function's behavior and identify the set of all possible output values. This may involve analyzing the function's graph, identifying any asymptotes, and determining the function's maximum and minimum values.