The Functions F F F And G G G Are Defined As F ( X ) = 6 X + 5 F(x) = 6x + 5 F ( X ) = 6 X + 5 And G ( X ) = 3 − 8 X G(x) = 3 - 8x G ( X ) = 3 − 8 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 ​ , And

Introduction

In this article, we will explore the functions $f$ and $g$, which are defined as $f(x) = 6x + 5$ and $g(x) = 3 - 8x$. We will examine the domain of each function individually, as well as the domain of their various compositions, including $f+g$, $f-g$, $fg$, $ff$, and $\frac{f}{g}$.

Domain of $f$ and $g$

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 valid.

Domain of $f$

The function $f(x) = 6x + 5$ is a linear function, which means that it is defined for all real numbers. Therefore, the domain of $f$ is the set of all real numbers, denoted by $\mathbb{R}$.

Domain of $g$

The function $g(x) = 3 - 8x$ is also a linear function, which means that it is defined for all real numbers. Therefore, the domain of $g$ is also the set of all real numbers, denoted by $\mathbb{R}$.

Domain of $f+g$

The function $f+g$ is defined as $(f+g)(x) = f(x) + g(x) = (6x + 5) + (3 - 8x) = -2x + 8$. Since this function is also a linear function, it is defined for all real numbers. Therefore, the domain of $f+g$ is the set of all real numbers, denoted by $\mathbb{R}$.

Domain of $f-g$

The function $f-g$ is defined as $(f-g)(x) = f(x) - g(x) = (6x + 5) - (3 - 8x) = 14x + 2$. Since this function is also a linear function, it is defined for all real numbers. Therefore, the domain of $f-g$ is the set of all real numbers, denoted by $\mathbb{R}$.

Domain of $fg$

The function $fg$ is defined as $(fg)(x) = f(x) \cdot g(x) = (6x + 5) \cdot (3 - 8x) = -48x^2 - 22x + 15$. Since this function is a quadratic function, it is defined for all real numbers. Therefore, the domain of $fg$ is the set of all real numbers, denoted by $\mathbb{R}$.

Domain of $ff$

The function $ff$ is defined as $(ff)(x) = f(f(x)) = f(6x + 5) = 6(6x + 5) + 5 = 36x + 35$. Since this function is a linear function, it is defined for all real numbers. Therefore, the domain of $ff$ is the set of all real numbers, denoted by $\mathbb{R}$.

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

The function $\frac{f}{g}$ is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{6x + 5}{3 - 8x}$. Since this function is a rational function, it is defined for all real numbers except for the values of $x$ that make the denominator equal to zero. Therefore, the domain of $\frac{f}{g}$ is the set of all real numbers except for the values of $x$ that satisfy the equation $3 - 8x = 0$, which is equivalent to $x = \frac{3}{8}$.

Conclusion

In conclusion, the domain of $f$ and $g$ is the set of all real numbers, denoted by $\mathbb{R}$. The domain of $f+g$, $f-g$, $fg$, and $ff$ is also the set of all real numbers, denoted by $\mathbb{R}$. However, the domain of $\frac{f}{g}$ is the set of all real numbers except for the value of $x$ that satisfies the equation $3 - 8x = 0$, which is equivalent to $x = \frac{3}{8}$.

References

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

Future Work

Q: What is the domain of the function $f(x) = 6x + 5$?

A: The domain of the function $f(x) = 6x + 5$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function is a linear function, which means that it is defined for all real numbers.

Q: What is the domain of the function $g(x) = 3 - 8x$?

A: The domain of the function $g(x) = 3 - 8x$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function is a linear function, which means that it is defined for all real numbers.

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

A: The domain of the function $f+g$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function $f+g$ is defined as $(f+g)(x) = f(x) + g(x) = (6x + 5) + (3 - 8x) = -2x + 8$, which is a linear function.

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

A: The domain of the function $f-g$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function $f-g$ is defined as $(f-g)(x) = f(x) - g(x) = (6x + 5) - (3 - 8x) = 14x + 2$, which is a linear function.

Q: What is the domain of the function $fg$?

A: The domain of the function $fg$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function $fg$ is defined as $(fg)(x) = f(x) \cdot g(x) = (6x + 5) \cdot (3 - 8x) = -48x^2 - 22x + 15$, which is a quadratic function.

Q: What is the domain of the function $ff$?

A: The domain of the function $ff$ is the set of all real numbers, denoted by $\mathbb{R}$. This is because the function $ff$ is defined as $(ff)(x) = f(f(x)) = f(6x + 5) = 6(6x + 5) + 5 = 36x + 35$, which is a linear function.

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

A: The domain of the function $\frac{f}{g}$ is the set of all real numbers except for the value of $x$ that satisfies the equation $3 - 8x = 0$, which is equivalent to $x = \frac{3}{8}$. This is because the function $\frac{f}{g}$ is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{6x + 5}{3 - 8x}$, which is a rational function.

Q: Why is the domain of the function $\frac{f}{g}$ different from the domain of the other functions?

A: The domain of the function $\frac{f}{g}$ is different from the domain of the other functions because the function $\frac{f}{g}$ is a rational function, which means that it is defined for all real numbers except for the values of $x$ that make the denominator equal to zero. In this case, the denominator is $3 - 8x$, which is equal to zero when $x = \frac{3}{8}$.

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. In other words, it determines the set of all possible values of $x$ for which the function is valid.

Q: How can I determine the domain of a function?

A: You can determine the domain of a function by examining the function's definition and identifying any restrictions on the input values. For example, if a function is defined as a rational function, you can determine its domain by identifying the values of $x$ that make the denominator equal to zero.