Let F ( X ) = 3 X − 2 F(x) = \sqrt{3x - 2} F ( X ) = 3 X − 2 ​ And G ( X ) = X G(x) = X G ( X ) = X . Find ( F + G ) ( X (f+g)(x ( F + G ) ( X ], ( F − G ) ( X (f-g)(x ( F − G ) ( X ], ( F G ) ( X (fg)(x ( F G ) ( X ], And F G ( X \frac{f}{g}(x G F ​ ( X ]. Give The Domain Of Each.1. ( F + G ) ( X ) = (f+g)(x) = ( F + G ) ( X ) = (Simplify Your Answer.)2. $(f-g)(x)

$$$$$$$$$$$$

Introduction

In this article, we will explore the concept of function composition and the domain of each composite function. We will start by defining two functions, $f(x) = \sqrt{3x - 2}$ and $g(x) = x$. Then, we will find the composite functions $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\frac{f}{g}(x)$. Finally, we will determine the domain of each composite function.

Finding the Composite Functions

To find the composite functions, we will use the following formulas:

• $(f+g)(x) = f(x) + g(x)$
• $(f-g)(x) = f(x) - g(x)$
• $(fg)(x) = f(x) \cdot g(x)$
• $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$

$(f+g)(x)$

To find $(f+g)(x)$, we will add $f(x)$ and $g(x)$.

$(f+g)(x) = f(x) + g(x)$ $= \sqrt{3x - 2} + x$

$(f-g)(x)$

To find $(f-g)(x)$, we will subtract $g(x)$ from $f(x)$.

$(f-g)(x) = f(x) - g(x)$ $= \sqrt{3x - 2} - x$

$(fg)(x)$

To find $(fg)(x)$, we will multiply $f(x)$ and $g(x)$.

$(fg)(x) = f(x) \cdot g(x)$ $= \sqrt{3x - 2} \cdot x$

$\frac{f}{g}(x)$

To find $\frac{f}{g}(x)$, we will divide $f(x)$ by $g(x)$.

$\frac{f}{g}(x) = \frac{f(x)}{g(x)}$ $= \frac{\sqrt{3x - 2}}{x}$

Domain of Each Composite Function

To determine the domain of each composite function, we need to consider the restrictions on the domain of each individual function.

Domain of $(f+g)(x)$

The domain of $(f+g)(x)$ is the set of all values of $x$ for which $f(x)$ and $g(x)$ are both defined.

Since $f(x) = \sqrt{3x - 2}$, we know that $3x - 2 \geq 0$, which implies that $x \geq \frac{2}{3}$.

Since $g(x) = x$, we know that $g(x)$ is defined for all real numbers.

Therefore, the domain of $(f+g)(x)$ is $x \geq \frac{2}{3}$.

Domain of $(f-g)(x)$

The domain of $(f-g)(x)$ is the set of all values of $x$ for which $f(x)$ and $g(x)$ are both defined.

Since $f(x) = \sqrt{3x - 2}$, we know that $3x - 2 \geq 0$, which implies that $x \geq \frac{2}{3}$.

Since $g(x) = x$, we know that $g(x)$ is defined for all real numbers.

Therefore, the domain of $(f-g)(x)$ is $x \geq \frac{2}{3}$.

Domain of $(fg)(x)$

The domain of $(fg)(x)$ is the set of all values of $x$ for which $f(x)$ and $g(x)$ are both defined.

Since $f(x) = \sqrt{3x - 2}$, we know that $3x - 2 \geq 0$, which implies that $x \geq \frac{2}{3}$.

Since $g(x) = x$, we know that $g(x)$ is defined for all real numbers.

Therefore, the domain of $(fg)(x)$ is $x \geq \frac{2}{3}$.

Domain of $\frac{f}{g}(x)$

The domain of $\frac{f}{g}(x)$ is the set of all values of $x$ for which $f(x)$ and $g(x)$ are both defined.

Since $f(x) = \sqrt{3x - 2}$, we know that $3x - 2 \geq 0$, which implies that $x \geq \frac{2}{3}$.

Since $g(x) = x$, we know that $g(x)$ is defined for all real numbers.

However, we also know that $g(x) \neq 0$, since we cannot divide by zero.

Therefore, the domain of $\frac{f}{g}(x)$ is $x > \frac{2}{3}$.

Conclusion

In this article, we have found the composite functions $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\frac{f}{g}(x)$, and determined the domain of each. We have shown that the domain of each composite function is $x \geq \frac{2}{3}$, except for $\frac{f}{g}(x)$, which has a domain of $x > \frac{2}{3}$.

Introduction

In our previous article, we explored the concept of composite functions and determined the domain of each composite function. In this article, we will answer some frequently asked questions about composite functions and domains.

Q: What is a composite function?

A: A composite function is a function that is formed by combining two or more functions. For example, if we have two functions $f(x)$ and $g(x)$, then the composite function $(f \circ g)(x)$ is defined as $f(g(x))$.

Q: How do I find the composite function $(f+g)(x)$?

A: To find the composite function $(f+g)(x)$, you need to add the two functions $f(x)$ and $g(x)$. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then $(f+g)(x) = \sqrt{3x - 2} + x$.

Q: How do I find the composite function $(f-g)(x)$?

A: To find the composite function $(f-g)(x)$, you need to subtract the two functions $f(x)$ and $g(x)$. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then $(f-g)(x) = \sqrt{3x - 2} - x$.

Q: How do I find the composite function $(fg)(x)$?

A: To find the composite function $(fg)(x)$, you need to multiply the two functions $f(x)$ and $g(x)$. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then $(fg)(x) = \sqrt{3x - 2} \cdot x$.

Q: How do I find the composite function $\frac{f}{g}(x)$?

A: To find the composite function $\frac{f}{g}(x)$, you need to divide the two functions $f(x)$ and $g(x)$. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then $\frac{f}{g}(x) = \frac{\sqrt{3x - 2}}{x}$.

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all values of $x$ for which the individual functions are both defined.

Q: How do I determine the domain of a composite function?

A: To determine the domain of a composite function, you need to consider the restrictions on the domain of each individual function. For example, if $f(x) = \sqrt{3x - 2}$, then the domain of $f(x)$ is $x \geq \frac{2}{3}$. If $g(x) = x$, then the domain of $g(x)$ is all real numbers.

Q: What is the difference between the domain of a composite function and the domain of an individual function?

A: The domain of a composite function is the set of all values of $x$ for which the individual functions are both defined. The domain of an individual function is the set of all values of $x$ for which the function is defined.

Q: Can a composite function have a different domain than the individual functions?

A: Yes, a composite function can have a different domain than the individual functions. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then the domain of $(f \circ g)(x)$ is $x > \frac{2}{3}$, which is different from the domain of $f(x)$ and $g(x)$.

Q: How do I graph a composite function?

A: To graph a composite function, you need to graph the individual functions and then combine them. For example, if $f(x) = \sqrt{3x - 2}$ and $g(x) = x$, then the graph of $(f \circ g)(x)$ is the graph of $f(x)$ shifted to the right by $\frac{2}{3}$.

Q: Can a composite function be a linear function?

A: Yes, a composite function can be a linear function. For example, if $f(x) = 2x$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a linear function.

Q: Can a composite function be a quadratic function?

A: Yes, a composite function can be a quadratic function. For example, if $f(x) = x^2$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a quadratic function.

Q: Can a composite function be a polynomial function?

A: Yes, a composite function can be a polynomial function. For example, if $f(x) = x^3$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a polynomial function.

Q: Can a composite function be a rational function?

A: Yes, a composite function can be a rational function. For example, if $f(x) = \frac{1}{x}$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a rational function.

Q: Can a composite function be a trigonometric function?

A: Yes, a composite function can be a trigonometric function. For example, if $f(x) = \sin x$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a trigonometric function.

Q: Can a composite function be an exponential function?

A: Yes, a composite function can be an exponential function. For example, if $f(x) = e^x$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is an exponential function.

Q: Can a composite function be a logarithmic function?

A: Yes, a composite function can be a logarithmic function. For example, if $f(x) = \log x$ and $g(x) = x$, then the composite function $(f \circ g)(x)$ is a logarithmic function.

Conclusion

In this article, we have answered some frequently asked questions about composite functions and domains. We have shown that composite functions can be used to create new functions from existing functions, and that the domain of a composite function is the set of all values of $x$ for which the individual functions are both defined. We have also shown that composite functions can be used to create a wide range of functions, including linear, quadratic, polynomial, rational, trigonometric, exponential, and logarithmic functions.