Given $f(x)=4x^2-5$ And \$g(x)=x+3$[/tex\].What Is $(f \cdot G)(x)$?

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When dealing with functions, it's not uncommon to encounter the task of multiplying two functions together. This process is known as function multiplication or composition. In this article, we will explore how to multiply two functions, using the given functions $f(x)=4x^2-5$ and $g(x)=x+3$ as examples.

Understanding Function Multiplication

Function multiplication is a process where we multiply two functions together to create a new function. This new function is a result of combining the two original functions. The process involves replacing the variable in one function with the other function, and then simplifying the resulting expression.

Step 1: Replace the Variable

To multiply two functions, we need to replace the variable in one function with the other function. In this case, we will replace the variable x in function f(x) with function g(x).

f(x)=4x2−5f(x) = 4x^2 - 5

g(x)=x+3g(x) = x + 3

We will replace x in function f(x) with g(x):

f(g(x))=4(g(x))2−5f(g(x)) = 4(g(x))^2 - 5

Step 2: Simplify the Expression

Now that we have replaced the variable, we need to simplify the resulting expression. To do this, we will expand the squared term and then combine like terms.

f(g(x))=4(g(x))2−5f(g(x)) = 4(g(x))^2 - 5

f(g(x))=4(x+3)2−5f(g(x)) = 4(x + 3)^2 - 5

To expand the squared term, we will use the formula (a + b)^2 = a^2 + 2ab + b^2.

f(g(x))=4(x2+6x+9)−5f(g(x)) = 4(x^2 + 6x + 9) - 5

Now, we will distribute the 4 to the terms inside the parentheses:

f(g(x))=4x2+24x+36−5f(g(x)) = 4x^2 + 24x + 36 - 5

Finally, we will combine like terms:

f(g(x))=4x2+24x+31f(g(x)) = 4x^2 + 24x + 31

Conclusion

In this article, we have explored how to multiply two functions together. We used the given functions $f(x)=4x^2-5$ and $g(x)=x+3$ as examples and walked through the step-by-step process of function multiplication. By replacing the variable in one function with the other function and simplifying the resulting expression, we arrived at the final function:

(fâ‹…g)(x)=4x2+24x+31(f \cdot g)(x) = 4x^2 + 24x + 31

This new function is a result of combining the two original functions and is a fundamental concept in mathematics.

Example Problems

To reinforce your understanding of function multiplication, try the following example problems:

  1. Multiply the functions $f(x) = 2x^2 - 3$ and $g(x) = x - 2$.
  2. Multiply the functions $f(x) = x^2 + 4x - 5$ and $g(x) = 2x + 1$.

Tips and Tricks

When multiplying two functions together, remember to:

  • Replace the variable in one function with the other function.
  • Simplify the resulting expression by expanding squared terms and combining like terms.
  • Use the formula (a + b)^2 = a^2 + 2ab + b^2 to expand squared terms.

By following these steps and tips, you will be able to multiply two functions together with ease and confidence.

Common Mistakes

When multiplying two functions together, be careful not to:

  • Forget to replace the variable in one function with the other function.
  • Simplify the resulting expression incorrectly.
  • Use the wrong formula to expand squared terms.

By avoiding these common mistakes, you will be able to multiply two functions together accurately and efficiently.

Real-World Applications

Function multiplication has many real-world applications in fields such as:

  • Physics: When dealing with complex systems, function multiplication can be used to model and analyze the behavior of the system.
  • Engineering: Function multiplication can be used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Function multiplication can be used to model and analyze economic systems, such as supply and demand curves.

By understanding function multiplication, you will be able to apply it to real-world problems and make informed decisions.

Conclusion

In the previous article, we explored the concept of function multiplication and walked through the step-by-step process of multiplying two functions together. In this article, we will answer some frequently asked questions about function multiplication to help you better understand this concept.

Q: What is function multiplication?

A: Function multiplication is a process where we multiply two functions together to create a new function. This new function is a result of combining the two original functions.

Q: How do I multiply two functions together?

A: To multiply two functions together, you need to replace the variable in one function with the other function and then simplify the resulting expression.

Q: What is the formula for function multiplication?

A: There is no specific formula for function multiplication. Instead, you need to follow the step-by-step process of replacing the variable and simplifying the resulting expression.

Q: Can I multiply more than two functions together?

A: Yes, you can multiply more than two functions together. However, the process becomes more complex and requires careful attention to detail.

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

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

  • Forgetting to replace the variable in one function with the other function.
  • Simplifying the resulting expression incorrectly.
  • Using the wrong formula to expand squared terms.

Q: How do I know when to use function multiplication?

A: You should use function multiplication when you need to combine two or more functions to create a new function. This is often the case in real-world applications such as physics, engineering, and economics.

Q: Can I use function multiplication to solve real-world problems?

A: Yes, function multiplication can be used to solve real-world problems in fields such as physics, engineering, and economics. By understanding function multiplication, you will be able to apply it to real-world problems and make informed decisions.

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

A: Some examples of real-world applications of function multiplication include:

  • Modeling and analyzing complex systems in physics.
  • Designing and optimizing systems in engineering.
  • Modeling and analyzing economic systems in economics.

Q: How do I practice function multiplication?

A: You can practice function multiplication by working through example problems and exercises. You can also try applying function multiplication to real-world problems to see how it can be used to solve complex problems.

Q: What are some resources for learning more about function multiplication?

A: Some resources for learning more about function multiplication include:

  • Online tutorials and videos.
  • Textbooks and reference books.
  • Online forums and discussion groups.

Conclusion

In conclusion, function multiplication is a fundamental concept in mathematics that involves multiplying two functions together to create a new function. By understanding function multiplication, you will be able to apply it to real-world problems and make informed decisions. Remember to avoid common mistakes and practice function multiplication to become proficient in this concept.

Additional Resources

For more information on function multiplication, check out the following resources:

  • Khan Academy: Function Multiplication
  • Mathway: Function Multiplication
  • Wolfram Alpha: Function Multiplication

Practice Problems

Try the following practice problems to test your understanding of function multiplication:

  1. Multiply the functions $f(x) = 2x^2 - 3$ and $g(x) = x - 2$.
  2. Multiply the functions $f(x) = x^2 + 4x - 5$ and $g(x) = 2x + 1$.
  3. Multiply the functions $f(x) = 3x^2 - 2x + 1$ and $g(x) = x^2 + 2x - 3$.

Answer Key

  1. (f⋅g)(x)=2x3−7x−6(f \cdot g)(x) = 2x^3 - 7x - 6

  2. (f⋅g)(x)=2x3+9x2−9x−5(f \cdot g)(x) = 2x^3 + 9x^2 - 9x - 5

  3. (f⋅g)(x)=9x4−4x3−5x2+6x−3(f \cdot g)(x) = 9x^4 - 4x^3 - 5x^2 + 6x - 3