Which Expression Is Equivalent To \[$(f+g)(4)\$\]?A. \[$f(4) + G(4)\$\]B. \[$f(x) + G(4)\$\]C. \[$f(4+g(4))\$\]D. \[$4(f(x) + G(x))\$\]

Understanding the Concept of Function Addition

In mathematics, when we add two functions, we create a new function that combines the properties of both functions. This new function is called the sum of the two original functions. The process of adding two functions is denoted by the symbol ${+}$ and is read as "the sum of ${f}$ and ${g}$" or ${f+g}$.

The Concept of Function Composition

Function composition is another fundamental concept in mathematics. It involves combining two or more functions to create a new function. When we compose two functions, we take the output of one function and use it as the input for the other function. This process is denoted by the symbol ${\circ}$ and is read as "the composition of ${f}$ and ${g}$" or ${f\circ g}$.

Understanding the Notation {(f+g)(4)$}$

The notation {(f+g)(4)$}$ represents the value of the function ${f+g}$ at the point ${x=4}$. In other words, it represents the value of the sum of the functions ${f}$ and ${g}$ at the point ${x=4}$.

Evaluating the Options

Now that we have a clear understanding of the concepts involved, let's evaluate the options given:

Option A: {f(4) + g(4)$]

This option represents the value of the function ${f}$ at the point ${x=4}$ added to the value of the function ${g}$ at the point ${x=4}$. This is a straightforward evaluation of the functions at the given point.

Option B: [$f(x) + g(4)$]

This option represents the value of the function ${f}$ at the point ${x}$ added to the value of the function ${g}$ at the point ${x=4}$. This option is incorrect because it does not evaluate the function ${f+g}$ at the point ${x=4}$.

Option C: [f(4+g(4))\$}

This option represents the value of the function ${f}$ at the point ${x=4+g(4)}$. This option is incorrect because it does not evaluate the function ${f+g}$ at the point ${x=4}$.

Option D: ${4(f(x) + g(x))\$}

This option represents the value of the function ${4(f(x) + g(x))}$ at the point ${x=4}$. This option is incorrect because it does not evaluate the function ${f+g}$ at the point ${x=4}$.

Conclusion

Based on the understanding of the concepts involved, the correct option is A. {f(4) + g(4)$}$. This option represents the value of the function ${f+g}$ at the point ${x=4}$, which is the correct interpretation of the notation {(f+g)(4)$}$.

Final Answer

The final answer is A. {f(4) + g(4)$}$.

Understanding Function Addition and Composition

In mathematics, function addition and composition are fundamental concepts that help us understand how to combine functions to create new functions. In this article, we will answer some frequently asked questions (FAQs) on function addition and composition.

Q: What is function addition?

A: Function addition is the process of combining two or more functions to create a new function. This new function is called the sum of the original functions. The process of adding two functions is denoted by the symbol ${+}$ and is read as "the sum of ${f}$ and ${g}$" or ${f+g}$.

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. When we compose two functions, we take the output of one function and use it as the input for the other function. This process is denoted by the symbol ${\circ}$ and is read as "the composition of ${f}$ and ${g}$" or ${f\circ g}$.

Q: How do I evaluate a function at a given point?

A: To evaluate a function at a given point, you need to substitute the value of the point into the function. For example, if we have a function ${f(x) = 2x + 3}$ and we want to evaluate it at the point ${x=4}$, we would substitute ${x=4}$ into the function to get ${f(4) = 2(4) + 3 = 11}$.

Q: What is the difference between function addition and function composition?

A: Function addition and function composition are two different concepts in mathematics. Function addition involves combining two or more functions to create a new function, while function composition involves combining two or more functions to create a new function by taking the output of one function and using it as the input for the other function.

Q: How do I determine if two functions are equivalent?

A: To determine if two functions are equivalent, you need to check if they have the same output for a given input. If the outputs are the same for all possible inputs, then the functions are equivalent.

Q: What is the notation {(f+g)(4)$}$?

A: The notation {(f+g)(4)$}$ represents the value of the function ${f+g}$ at the point ${x=4}$. In other words, it represents the value of the sum of the functions ${f}$ and ${g}$ at the point ${x=4}$.

Q: How do I evaluate the notation {(f+g)(4)$}$?

A: To evaluate the notation {(f+g)(4)$}$, you need to substitute the value of the point ${x=4}$ into the function ${f+g}$. This means you need to add the values of the functions ${f}$ and ${g}$ at the point ${x=4}$.

Q: What is the final answer to the expression {(f+g)(4)$}$?

A: The final answer to the expression {(f+g)(4)$}$ is A. {f(4) + g(4)$}$. This option represents the value of the function ${f+g}$ at the point ${x=4}$, which is the correct interpretation of the notation {(f+g)(4)$}$.

Q: What are some common mistakes to avoid when working with function addition and composition?

A: Some common mistakes to avoid when working with function addition and composition include:

• Confusing function addition with function composition
• Not evaluating functions at the correct point
• Not checking if two functions are equivalent
• Not using the correct notation for function addition and composition

Q: How can I practice function addition and composition?

A: You can practice function addition and composition by working through examples and exercises. You can also try creating your own functions and evaluating them at different points. Additionally, you can use online resources and tools to help you practice and learn.

Q: What are some real-world applications of function addition and composition?

A: Function addition and composition have many real-world applications, including:

• Physics: Function composition is used to describe the motion of objects in terms of their position, velocity, and acceleration.
• Engineering: Function addition and composition are used to design and analyze complex systems, such as electrical circuits and mechanical systems.
• Economics: Function composition is used to model economic systems and make predictions about future economic trends.

Q: How can I learn more about function addition and composition?

A: You can learn more about function addition and composition by:

• Reading textbooks and online resources
• Working through examples and exercises
• Practicing with real-world applications
• Seeking help from teachers or tutors
• Joining online communities and forums to discuss function addition and composition with others.