If $f(x)=4^x-8$ And $g(x)=5x+6$, Find $(f+g)(x$\].A. $(f+g)(x)=4^x-3x+6$ B. $(f+g)(x)=9x-2$ C. $(f+g)(x)=4^x+5x-2$ D. $(f+g)(x)=-4^x-5x+2$

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Understanding the Problem

To find the value of $(f+g)(x)$, we need to understand what the notation means. The expression $(f+g)(x)$ represents the sum of two functions, $f(x)$ and $g(x)$, evaluated at a specific value of $x$. In other words, we need to add the two functions together, term by term, and simplify the resulting expression.

Step 1: Write Down the Functions

The two functions are given as:

$f(x) = 4^x - 8$

$g(x) = 5x + 6$

Step 2: Add the Functions

To add the functions, we need to add the corresponding terms. In this case, we have:

$(f+g)(x) = (4^x - 8) + (5x + 6)$

Step 3: Simplify the Expression

Now, we need to simplify the expression by combining like terms. We can do this by adding the coefficients of the same variables:

$(f+g)(x) = 4^x + 5x - 8 + 6$

Step 4: Combine Like Terms

We can combine the constant terms by adding them together:

$(f+g)(x) = 4^x + 5x - 2$

Conclusion

Therefore, the value of $(f+g)(x)$ is:

$(f+g)(x) = 4^x + 5x - 2$

This is option C in the given choices.

Why is This Important?

Understanding how to add functions is an important concept in mathematics, particularly in calculus and algebra. It allows us to combine different functions to create new ones, which can be used to model real-world phenomena. In this case, we used the concept of function addition to find the value of $(f+g)(x)$, which is a fundamental operation in mathematics.

Real-World Applications

The concept of function addition has many real-world applications, such as:

• Modeling population growth: By adding two functions that represent the growth rates of two different populations, we can create a new function that represents the total population growth.
• Analyzing financial data: By adding two functions that represent the income and expenses of a company, we can create a new function that represents the net income.
• Predicting weather patterns: By adding two functions that represent the temperature and humidity levels, we can create a new function that represents the overall weather conditions.

Common Mistakes

When adding functions, it's easy to make mistakes, such as:

• Forgetting to combine like terms
• Adding the wrong terms together
• Not simplifying the expression enough

To avoid these mistakes, it's essential to carefully read the problem, write down the functions, and simplify the expression step by step.

Tips and Tricks

Here are some tips and tricks to help you add functions like a pro:

• Make sure to write down the functions clearly and accurately
• Use a consistent notation for the variables and functions
• Combine like terms carefully and simplify the expression step by step
• Check your work by plugging in a few values to see if the expression holds true

By following these tips and tricks, you'll be able to add functions like a pro and solve problems with ease.

Conclusion

In conclusion, adding functions is a fundamental concept in mathematics that has many real-world applications. By understanding how to add functions, we can create new functions that can be used to model real-world phenomena. In this article, we used the concept of function addition to find the value of $(f+g)(x)$, which is a fundamental operation in mathematics. We also discussed the importance of function addition, real-world applications, common mistakes, and tips and tricks to help you add functions like a pro.

Understanding Function Addition

Adding functions is a fundamental concept in mathematics that has many real-world applications. In this article, we'll answer some common questions about adding functions and provide tips and tricks to help you master this concept.

Q: What is function addition?

A: Function addition is the process of combining two or more functions to create a new function. This is done by adding the corresponding terms of the functions together.

Q: How do I add functions?

A: To add functions, you need to follow these steps:

1. Write down the functions clearly and accurately
2. Combine like terms by adding the coefficients of the same variables
3. Simplify the expression by combining constant terms
4. Check your work by plugging in a few values to see if the expression holds true

Q: What are some common mistakes to avoid when adding functions?

A: Some common mistakes to avoid when adding functions include:

• Forgetting to combine like terms
• Adding the wrong terms together
• Not simplifying the expression enough
• Not checking your work by plugging in a few values

Q: How do I know if I've added functions correctly?

A: To check if you've added functions correctly, you can plug in a few values to see if the expression holds true. You can also use a calculator or computer software to check your work.

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

A: Function addition has many real-world applications, including:

• Modeling population growth
• Analyzing financial data
• Predicting weather patterns
• Creating mathematical models for scientific and engineering problems

Q: Can I add more than two functions together?

A: Yes, you can add more than two functions together. However, you need to follow the same steps as before, combining like terms and simplifying the expression.

Q: How do I handle functions with different variables?

A: When adding functions with different variables, you need to use a common variable for all the functions. You can then combine like terms and simplify the expression.

Q: Can I add functions with different bases?

A: Yes, you can add functions with different bases. However, you need to use a common base for all the functions. You can then combine like terms and simplify the expression.

Q: How do I handle functions with exponents?

A: When adding functions with exponents, you need to use the rules of exponents to simplify the expression. You can then combine like terms and simplify the expression.

Q: Can I add functions with fractions?

A: Yes, you can add functions with fractions. However, you need to use a common denominator for all the fractions. You can then combine like terms and simplify the expression.

Conclusion

In conclusion, adding functions is a fundamental concept in mathematics that has many real-world applications. By understanding how to add functions, you can create new functions that can be used to model real-world phenomena. We hope this Q&A article has helped you understand function addition and provided you with the tips and tricks you need to master this concept.

Additional Resources

If you're looking for more information on function addition, here are some additional resources you can check out:

• Khan Academy: Function Addition
• Mathway: Function Addition
• Wolfram Alpha: Function Addition
• MIT OpenCourseWare: Function Addition

Practice Problems

Here are some practice problems to help you master function addition:

• Add the functions f(x) = 2x + 3 and g(x) = x - 2
• Add the functions f(x) = x^2 + 2x and g(x) = x^2 - 3x
• Add the functions f(x) = 3x + 2 and g(x) = 2x - 1

Conclusion

We hope this Q&A article has helped you understand function addition and provided you with the tips and tricks you need to master this concept. Remember to practice regularly and seek help when you need it. Good luck!