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

When dealing with functions, it's essential to understand how to evaluate and compose them correctly. In this article, we'll explore the concept of function composition and evaluate the given expression $(f+g)(4)$ to determine which of the provided options is equivalent.

What is Function Composition?

Function composition is the process of combining two or more functions to create a new function. This is achieved by applying the output of one function as the input to another function. In mathematical notation, function composition is represented as $(f \circ g)(x) = f(g(x))$.

Evaluating the Expression $(f+g)(4)$

Given the expression $(f+g)(4)$, we need to determine which of the provided options is equivalent. To do this, we'll start by understanding what the expression means.

The expression $(f+g)(4)$ represents the composition of two functions, $f$ and $g$, evaluated at the point $x=4$. The $+$ symbol between $f$ and $g$ indicates that we're adding the two functions together, resulting in a new function.

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

Option A suggests that the expression $(f+g)(4)$ is equivalent to $f(4) + g(4)$. This option is tempting, as it seems to follow the order of operations. However, we need to consider the composition of the functions.

When we add two functions together, we're creating a new function that takes an input $x$ and returns the sum of the outputs of the two individual functions. In this case, the new function is $(f+g)(x) = f(x) + g(x)$.

Evaluating this new function at the point $x=4$, we get $(f+g)(4) = f(4) + g(4)$. However, this is not the only possibility.

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

Option B suggests that the expression $(f+g)(4)$ is equivalent to $f(x) + g(4)$. This option is incorrect, as it's not evaluating the new function at the point $x=4$. Instead, it's evaluating the function $f$ at the variable $x$ and adding the output of the function $g$ evaluated at the point $x=4$.

Option C: $f(4) + g(4)$

Option C is identical to Option A, suggesting that the expression $(f+g)(4)$ is equivalent to $f(4) + g(4)$. As we discussed earlier, this option is correct, as it's evaluating the new function at the point $x=4$.

Option D: $4(f(x) + g(x))$

Option D suggests that the expression $(f+g)(4)$ is equivalent to $4(f(x) + g(x))$. This option is incorrect, as it's not evaluating the new function at the point $x=4$. Instead, it's multiplying the new function by the constant $4$.

Conclusion

In conclusion, the correct answer is Option C: $f(4) + g(4)$. This option correctly evaluates the new function at the point $x=4$, resulting in the sum of the outputs of the two individual functions.

Understanding Function Composition and Evaluation: Key Takeaways

• Function composition is the process of combining two or more functions to create a new function.
• When evaluating an expression like $(f+g)(4)$, we need to consider the composition of the functions.
• The correct answer is Option C: $f(4) + g(4)$, as it evaluates the new function at the point $x=4$.

Additional Resources

For further reading on function composition and evaluation, we recommend the following resources:

• Khan Academy: Function Composition
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: Function Composition

In our previous article, we explored the concept of function composition and evaluated the expression $(f+g)(4)$ to determine which of the provided options is equivalent. In this article, we'll answer some frequently asked questions about function composition and evaluation.

Q: What is function composition?

A: Function composition is the process of combining two or more functions to create a new function. This is achieved by applying the output of one function as the input to another function. In mathematical notation, function composition is represented as $(f \circ g)(x) = f(g(x))$.

Q: How do I evaluate an expression like $(f+g)(4)$?

A: To evaluate an expression like $(f+g)(4)$, you need to consider the composition of the functions. In this case, the new function is $(f+g)(x) = f(x) + g(x)$. Evaluating this new function at the point $x=4$, we get $(f+g)(4) = f(4) + g(4)$.

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

A: Function composition and function addition are two different operations. Function composition involves combining two or more functions to create a new function, while function addition involves adding two or more functions together. In the case of $(f+g)(4)$, we're adding the two functions together, resulting in a new function.

Q: Can I use function composition to simplify complex expressions?

A: Yes, function composition can be used to simplify complex expressions. By breaking down the expression into smaller components and applying function composition, you can simplify the expression and make it easier to evaluate.

Q: How do I know which option is correct when evaluating an expression like $(f+g)(4)$?

A: To determine which option is correct, you need to evaluate the expression and compare it to the options provided. In the case of $(f+g)(4)$, the correct answer is Option C: $f(4) + g(4)$.

Q: What are some common mistakes to avoid when evaluating expressions like $(f+g)(4)$?

A: Some common mistakes to avoid when evaluating expressions like $(f+g)(4)$ include:

• Not considering the composition of the functions
• Evaluating the expression at the wrong point
• Not adding the functions together correctly
• Not using the correct notation for function composition

Q: How can I practice function composition and evaluation?

A: You can practice function composition and evaluation by working through examples and exercises. Some resources to help you get started include:

• Khan Academy: Function Composition
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: Function Composition

Conclusion

In conclusion, function composition and evaluation are essential concepts in mathematics. By understanding how to evaluate expressions like $(f+g)(4)$ and avoiding common mistakes, you'll be better equipped to tackle complex mathematical problems and develop a deeper appreciation for the beauty of mathematics.

Additional Resources

For further reading on function composition and evaluation, we recommend the following resources:

• Khan Academy: Function Composition
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: Function Composition

By practicing function composition and evaluation, you'll become more confident and proficient in your ability to tackle complex mathematical problems.