Let $f(x)=\sqrt{3x-2}$ And $g(x)=\frac{1}{x}$.Find $ ( F + G ) ( X ) (f+g)(x) ( F + G ) ( X ) [/tex], $(f-g)(x)$, $(fg)(x)$, And $\left(\frac{f}{g}\right)(x)$. Give The Domain Of Each.1. $(f+g)(x)

Introduction

In mathematics, functions are used to describe relationships between variables. When we have two functions, we can combine them using various operations such as addition, subtraction, multiplication, and division. In this article, we will explore how to combine two given functions, $f(x)=\sqrt{3x-2}$ and $g(x)=\frac{1}{x}$, using these operations and determine the domain of each resulting function.

Combining Functions: Addition

To find the sum of two functions, we add their corresponding values. In this case, we need to find $(f+g)(x)$, which is the sum of $f(x)$ and $g(x)$.

$$(f+g)(x) = f(x) + g(x) = \sqrt{3x-2} + \frac{1}{x}$$

To determine the domain of $(f+g)(x)$, we need to consider the restrictions on both $f(x)$ and $g(x)$. The function $f(x)$ is defined only when $3x-2 \geq 0$, which implies $x \geq \frac{2}{3}$. The function $g(x)$ is defined only when $x \neq 0$. Therefore, the domain of $(f+g)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$.

Combining Functions: Subtraction

To find the difference of two functions, we subtract the corresponding values. In this case, we need to find $(f-g)(x)$, which is the difference of $f(x)$ and $g(x)$.

$$(f-g)(x) = f(x) - g(x) = \sqrt{3x-2} - \frac{1}{x}$$

To determine the domain of $(f-g)(x)$, we need to consider the restrictions on both $f(x)$ and $g(x)$. The function $f(x)$ is defined only when $3x-2 \geq 0$, which implies $x \geq \frac{2}{3}$. The function $g(x)$ is defined only when $x \neq 0$. Therefore, the domain of $(f-g)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$.

Combining Functions: Multiplication

To find the product of two functions, we multiply their corresponding values. In this case, we need to find $(fg)(x)$, which is the product of $f(x)$ and $g(x)$.

$$(fg)(x) = f(x) \cdot g(x) = \sqrt{3x-2} \cdot \frac{1}{x} = \frac{\sqrt{3x-2}}{x}$$

To determine the domain of $(fg)(x)$, we need to consider the restrictions on both $f(x)$ and $g(x)$. The function $f(x)$ is defined only when $3x-2 \geq 0$, which implies $x \geq \frac{2}{3}$. The function $g(x)$ is defined only when $x \neq 0$. Therefore, the domain of $(fg)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$.

Combining Functions: Division

To find the quotient of two functions, we divide the corresponding values. In this case, we need to find $\left(\frac{f}{g}\right)(x)$, which is the quotient of $f(x)$ and $g(x)$.

$$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{3x-2}}{\frac{1}{x}} = \sqrt{3x-2} \cdot x = x\sqrt{3x-2}$$

To determine the domain of $\left(\frac{f}{g}\right)(x)$, we need to consider the restrictions on both $f(x)$ and $g(x)$. The function $f(x)$ is defined only when $3x-2 \geq 0$, which implies $x \geq \frac{2}{3}$. The function $g(x)$ is defined only when $x \neq 0$. Therefore, the domain of $\left(\frac{f}{g}\right)(x)$ is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$.

Conclusion

In this article, we have combined two functions, $f(x)=\sqrt{3x-2}$ and $g(x)=\frac{1}{x}$, using addition, subtraction, multiplication, and division. We have determined the domain of each resulting function and found that the domain of each is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$.

Introduction

In our previous article, we explored how to combine two functions, $f(x)=\sqrt{3x-2}$ and $g(x)=\frac{1}{x}$, using addition, subtraction, multiplication, and division. We determined the domain of each resulting function and found that the domain of each is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq \frac{2}{3}$ and $x \neq 0$. In this article, we will answer some frequently asked questions about combining functions.

Q: What is the difference between combining functions and composing functions?

A: Combining functions involves adding, subtracting, multiplying, or dividing two functions, while composing functions involves substituting one function into another. For example, if we have two functions $f(x)$ and $g(x)$, we can combine them by adding $f(x)$ and $g(x)$, or we can compose them by substituting $g(x)$ into $f(x)$.

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

A: To determine the domain of a combined function, you need to consider the restrictions on both functions being combined. For example, if we have two functions $f(x)$ and $g(x)$, and $f(x)$ is defined only when $x \geq 2$, and $g(x)$ is defined only when $x \neq 0$, then the domain of the combined function is the intersection of the domains of $f(x)$ and $g(x)$, which is $x \geq 2$ and $x \neq 0$.

Q: Can I combine functions that have different domains?

A: Yes, you can combine functions that have different domains. However, you need to consider the restrictions on both functions being combined. For example, if we have two functions $f(x)$ and $g(x)$, and $f(x)$ is defined only when $x \geq 2$, and $g(x)$ is defined only when $x \leq -2$, then the domain of the combined function is the intersection of the domains of $f(x)$ and $g(x)$, which is empty.

Q: How do I simplify a combined function?

A: To simplify a combined function, you can use algebraic manipulations such as combining like terms, factoring, or canceling out common factors. For example, if we have a combined function $f(x) + g(x) = 2x + 3 + \frac{1}{x}$, we can simplify it by combining like terms to get $f(x) + g(x) = \frac{2x^2 + 3x + 1}{x}$.

Q: Can I combine functions that have different variables?

A: No, you cannot combine functions that have different variables. For example, if we have two functions $f(x)$ and $g(y)$, we cannot combine them by adding or subtracting them, because they have different variables.

Q: How do I determine the range of a combined function?

A: To determine the range of a combined function, you need to consider the range of both functions being combined. For example, if we have two functions $f(x)$ and $g(x)$, and $f(x)$ has a range of $[0, \infty)$, and $g(x)$ has a range of $(-\infty, 0]$, then the range of the combined function is the intersection of the ranges of $f(x)$ and $g(x)$, which is empty.

Conclusion

In this article, we have answered some frequently asked questions about combining functions. We have discussed how to determine the domain of a combined function, how to simplify a combined function, and how to determine the range of a combined function. We have also discussed the differences between combining functions and composing functions, and how to combine functions that have different domains or variables.