Let F ( X ) = X + 4 F(x) = X + 4 F ( X ) = X + 4 And G ( X ) = 3 X + 2 G(x) = 3x + 2 G ( X ) = 3 X + 2 . Find ( F − G ) ( X (f-g)(x ( F − G ) ( X ] And Use Interval Notation For Its Domain.$(f-g)(x) = $Domain:

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Introduction

In mathematics, functions are used to describe the relationship between variables. When we have two functions, $f(x)$ and $g(x)$, we can perform various operations on them, such as addition, subtraction, multiplication, and division. In this article, we will focus on finding the difference between two functions, $(f-g)(x)$, and determining its domain using interval notation.

Defining the Functions

Let's start by defining the two functions:

• $f(x) = x + 4$
• $g(x) = 3x + 2$

These functions are linear, meaning they have a constant slope. The function $f(x)$ has a slope of 1 and a y-intercept of 4, while the function $g(x)$ has a slope of 3 and a y-intercept of 2.

Finding the Difference

To find the difference between the two functions, we need to subtract $g(x)$ from $f(x)$. This is denoted as $(f-g)(x)$.

$(f-g)(x) = f(x) - g(x)$

Substituting the expressions for $f(x)$ and $g(x)$, we get:

$(f-g)(x) = (x + 4) - (3x + 2)$

Expanding and simplifying the expression, we get:

$(f-g)(x) = x + 4 - 3x - 2$

Combine like terms:

$(f-g)(x) = -2x + 2$

Domain of the Difference Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of the difference function $(f-g)(x)$, the domain is the set of all real numbers, since there are no restrictions on the input values.

However, we need to consider the domain of the original functions $f(x)$ and $g(x)$. Since both functions are linear, their domains are all real numbers.

Interval Notation for the Domain

To express the domain of the difference function in interval notation, we use the following notation:

$(-\infty, \infty)$

This notation indicates that the domain of the difference function is all real numbers, from negative infinity to positive infinity.

Conclusion

In conclusion, we have found the difference between the two functions $f(x)$ and $g(x)$, which is $(f-g)(x) = -2x + 2$. We have also determined the domain of the difference function using interval notation, which is $(-\infty, \infty)$. This indicates that the domain of the difference function is all real numbers.

Example Problems

Problem 1

Find the difference between the two functions $f(x) = 2x + 1$ and $g(x) = x - 3$.

Solution

To find the difference between the two functions, we need to subtract $g(x)$ from $f(x)$.

$(f-g)(x) = f(x) - g(x)$

Substituting the expressions for $f(x)$ and $g(x)$, we get:

$(f-g)(x) = (2x + 1) - (x - 3)$

Expanding and simplifying the expression, we get:

$(f-g)(x) = 2x + 1 - x + 3$

Combine like terms:

$(f-g)(x) = x + 4$

Problem 2

Find the domain of the difference function $(f-g)(x) = 3x - 2$.

Solution

The domain of the difference function is the set of all real numbers, since there are no restrictions on the input values.

Therefore, the domain of the difference function is:

$(-\infty, \infty)$

Final Thoughts

In this article, we have found the difference between two functions, $(f-g)(x)$, and determined its domain using interval notation. We have also provided example problems to illustrate the concept. The difference function is an important concept in mathematics, and it has many applications in various fields, such as physics, engineering, and economics.

Introduction

In our previous article, we discussed how to find the difference between two functions, $(f-g)(x)$, and determined its domain using interval notation. In this article, we will answer some frequently asked questions about finding the difference between two functions.

Q1: What is the difference between two functions?

A1: The difference between two functions, $(f-g)(x)$, is a new function that is obtained by subtracting one function from another. It is denoted as $(f-g)(x)$ and is equal to $f(x) - g(x)$.

Q2: How do I find the difference between two functions?

A2: To find the difference between two functions, you need to subtract one function from another. This is denoted as $(f-g)(x)$ and is equal to $f(x) - g(x)$. You can substitute the expressions for $f(x)$ and $g(x)$ and simplify the resulting expression.

Q3: What is the domain of the difference function?

A3: The domain of the difference function is the set of all real numbers, since there are no restrictions on the input values. However, you need to consider the domain of the original functions $f(x)$ and $g(x)$. If either of these functions has a restricted domain, then the domain of the difference function will also be restricted.

Q4: How do I express the domain of the difference function in interval notation?

A4: To express the domain of the difference function in interval notation, you use the following notation:

$(-\infty, \infty)$

This notation indicates that the domain of the difference function is all real numbers, from negative infinity to positive infinity.

Q5: Can I find the difference between two functions that have different domains?

A5: Yes, you can find the difference between two functions that have different domains. However, you need to consider the domain of the original functions $f(x)$ and $g(x)$. If either of these functions has a restricted domain, then the domain of the difference function will also be restricted.

Q6: How do I find the difference between two functions that have the same domain?

A6: If two functions have the same domain, then the domain of the difference function will also be the same. You can simply subtract one function from another to find the difference function.

Q7: Can I find the difference between two functions that are not linear?

A7: Yes, you can find the difference between two functions that are not linear. However, you need to be careful when simplifying the resulting expression, as it may involve more complex algebraic manipulations.

Q8: How do I determine the domain of the difference function when the original functions have a restricted domain?

A8: To determine the domain of the difference function when the original functions have a restricted domain, you need to consider the intersection of the domains of the two functions. This will give you the domain of the difference function.

Q9: Can I find the difference between two functions that have a restricted domain and an infinite domain?

A9: Yes, you can find the difference between two functions that have a restricted domain and an infinite domain. However, you need to be careful when simplifying the resulting expression, as it may involve more complex algebraic manipulations.

Q10: How do I express the domain of the difference function in interval notation when the original functions have a restricted domain?

A10: To express the domain of the difference function in interval notation when the original functions have a restricted domain, you need to consider the intersection of the domains of the two functions. This will give you the domain of the difference function, which you can then express in interval notation.

Conclusion

In conclusion, finding the difference between two functions is an important concept in mathematics. By understanding how to find the difference between two functions and determining its domain using interval notation, you can apply this concept to a wide range of problems in various fields.