Suppose That The Functions $f$ And $g$ Are Defined For All Real Numbers $x$ As Follows:$ \begin{array}{l} f(x)=x+1 \\ g(x)=3x+1 \end{array} $Write The Expressions For $(f-g)(x$\] And $(f \cdot

Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows:

Introduction

In mathematics, functions are used to describe the relationship between variables. Given two functions $f$ and $g$, we can perform various operations on them to obtain new functions. In this article, we will explore the expressions for $(f-g)(x)$ and $(f \cdot g)(x)$, where $f(x) = x + 1$ and $g(x) = 3x + 1$.

The Difference of Functions

The difference of two functions $f$ and $g$ is defined as $(f - g)(x) = f(x) - g(x)$. To find the expression for $(f - g)(x)$, we need to substitute the given expressions for $f(x)$ and $g(x)$ into the definition.

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

Therefore, the expression for $(f - g)(x)$ is $-2x$.

The Product of Functions

The product of two functions $f$ and $g$ is defined as $(f \cdot g)(x) = f(x) \cdot g(x)$. To find the expression for $(f \cdot g)(x)$, we need to substitute the given expressions for $f(x)$ and $g(x)$ into the definition.

(f \cdot g)(x) = f(x) \cdot g(x)
= (x + 1) \cdot (3x + 1)
= 3x^2 + x + 3x + 1
= 3x^2 + 4x + 1

Therefore, the expression for $(f \cdot g)(x)$ is $3x^2 + 4x + 1$.

Conclusion

In this article, we have derived the expressions for $(f - g)(x)$ and $(f \cdot g)(x)$, where $f(x) = x + 1$ and $g(x) = 3x + 1$. The difference of functions is defined as $(f - g)(x) = f(x) - g(x)$, and the product of functions is defined as $(f \cdot g)(x) = f(x) \cdot g(x)$. By substituting the given expressions for $f(x)$ and $g(x)$ into these definitions, we obtained the expressions $-2x$ and $3x^2 + 4x + 1$, respectively.

Example Use Cases

The expressions for $(f - g)(x)$ and $(f \cdot g)(x)$ have various applications in mathematics and other fields. For instance, in calculus, the difference of functions is used to find the derivative of a function, while the product of functions is used to find the integral of a function. In physics, the difference of functions is used to describe the motion of an object, while the product of functions is used to describe the force acting on an object.

Further Reading

For further reading on functions and their operations, we recommend the following resources:

• Wikipedia: Function (mathematics)
• Khan Academy: Functions
• MIT OpenCourseWare: Functions

References

• [1] "Functions" by Michael Artin, in Algebra (Prentice Hall, 1991)
• [2] "Calculus" by Michael Spivak, in Calculus (Benjamin/Cummings, 1980)
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler, in Physics for Scientists and Engineers (W.H. Freeman and Company, 1990)

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of sources on the topic.
Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows:

Q&A: Functions and Their Operations

In this article, we will answer some frequently asked questions about functions and their operations.

Q: What is the difference between a function and an equation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An equation, on the other hand, is a statement that two expressions are equal. For example, the equation $x + 1 = 3x + 1$ is not a function, but the expression $f(x) = x + 1$ is a function.

Q: How do you find the difference of two functions?

A: To find the difference of two functions $f$ and $g$, you need to subtract the second function from the first function. In other words, $(f - g)(x) = f(x) - g(x)$.

Q: How do you find the product of two functions?

A: To find the product of two functions $f$ and $g$, you need to multiply the two functions together. In other words, $(f \cdot g)(x) = f(x) \cdot g(x)$.

Q: What is the difference between the difference of functions and the product of functions?

A: The difference of functions is a new function that is obtained by subtracting one function from another. The product of functions is a new function that is obtained by multiplying two functions together.

Q: Can you give an example of how to find the difference of two functions?

A: Yes, let's say we have two functions $f(x) = x + 1$ and $g(x) = 3x + 1$. To find the difference of these two functions, we need to subtract $g(x)$ from $f(x)$. This gives us $(f - g)(x) = f(x) - g(x) = (x + 1) - (3x + 1) = -2x$.

Q: Can you give an example of how to find the product of two functions?

A: Yes, let's say we have two functions $f(x) = x + 1$ and $g(x) = 3x + 1$. To find the product of these two functions, we need to multiply $f(x)$ and $g(x)$ together. This gives us $(f \cdot g)(x) = f(x) \cdot g(x) = (x + 1) \cdot (3x + 1) = 3x^2 + 4x + 1$.

Q: What are some common applications of functions and their operations?

A: Functions and their operations have many applications in mathematics, science, and engineering. Some common applications include:

• Calculus: Functions and their operations are used to find derivatives and integrals.
• Physics: Functions and their operations are used to describe the motion of objects and the forces acting on them.
• Computer Science: Functions and their operations are used to write algorithms and programs.

Conclusion

In this article, we have answered some frequently asked questions about functions and their operations. We have also provided examples of how to find the difference of two functions and the product of two functions. We hope that this article has been helpful in understanding functions and their operations.

Further Reading

For further reading on functions and their operations, we recommend the following resources:

• Wikipedia: Function (mathematics)
• Khan Academy: Functions
• MIT OpenCourseWare: Functions

References

• [1] "Functions" by Michael Artin, in Algebra (Prentice Hall, 1991)
• [2] "Calculus" by Michael Spivak, in Calculus (Benjamin/Cummings, 1980)
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler, in Physics for Scientists and Engineers (W.H. Freeman and Company, 1990)

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of sources on the topic.