Finding the Product of Functions

Oct 13, 2025 by windailys.com 17 views

Introduction

In mathematics, functions are used to represent relationships between variables. When we are given two functions, we can perform various operations on them to create new functions. In this article, we will learn how to perform the indicated operation with the given functions and find the resulting function.

Given Functions

We are given two functions:

g(x) = 4x - 1
f(x) = x^2 - 2x

The Indicated Operation

The indicated operation is to find the product of the two functions, denoted as (g ⋅ f)(x).

Finding the Product of Functions

To find the product of two functions, we multiply the two functions together. This means that we multiply each term in the first function by each term in the second function.

Step 1: Multiply the First Function by the Second Function

We will multiply the first function, g(x) = 4x - 1, by the second function, f(x) = x^2 - 2x.

(g ⋅ f)(x) = (4x - 1)(x^2 - 2x)

Step 2: Expand the Product

To expand the product, we will use the distributive property, which states that we can multiply each term in the first function by each term in the second function.

(g ⋅ f)(x) = 4x(x^2 - 2x) - 1(x^2 - 2x)

Step 3: Simplify the Expression

Now, we will simplify the expression by multiplying the terms.

(g ⋅ f)(x) = 4x^3 - 8x^2 - x^2 + 2x

Step 4: Combine Like Terms

Finally, we will combine like terms to get the final expression.

(g ⋅ f)(x) = 4x^3 - 9x^2 + 2x

Conclusion

In this article, we learned how to perform the indicated operation with the given functions and find the resulting function. We multiplied the two functions together, expanded the product, simplified the expression, and combined like terms to get the final expression.

Answer

The final answer is:

(g ⋅ f)(x) = 4x^3 - 9x^2 + 2x

This is option A.

Discussion

The product of two functions is a new function that is created by multiplying the two functions together. This new function can be used to represent a new relationship between variables.

Example Use Case

Suppose we have a function that represents the cost of producing a certain product, and we want to find the cost of producing a certain quantity of the product. We can use the product of two functions to represent the cost of producing the product.

Conclusion

In conclusion, performing operations with given functions is an important concept in mathematics. By understanding how to perform these operations, we can create new functions that can be used to represent relationships between variables.

References

[1] "Functions" by Khan Academy
[2] "Multiplying Functions" by Mathway

FAQs

Q: What is the product of two functions? A: The product of two functions is a new function that is created by multiplying the two functions together.
Q: How do I find the product of two functions? A: To find the product of two functions, you multiply the two functions together, expand the product, simplify the expression, and combine like terms.
Q: What is the final answer? A: The final answer is (g ⋅ f)(x) = 4x^3 - 9x^2 + 2x.
Performing Operations with Given Functions: Q&A =============================================

Introduction

In our previous article, we learned how to perform the indicated operation with the given functions and find the resulting function. In this article, we will answer some frequently asked questions (FAQs) related to performing operations with given functions.

Q&A

Q: What is the product of two functions?

A: The product of two functions is a new function that is created by multiplying the two functions together.

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

A: To find the product of two functions, you multiply the two functions together, expand the product, simplify the expression, and combine like terms.

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

A: The product of two functions is a new function that is created by multiplying the two functions together, while the composition of two functions is a new function that is created by plugging one function into another.

Q: Can I add or subtract functions?

A: Yes, you can add or subtract functions by combining like terms.

Q: How do I divide functions?

A: To divide functions, you can multiply the first function by the reciprocal of the second function.

Q: What is the final answer?

A: The final answer is (g ⋅ f)(x) = 4x^3 - 9x^2 + 2x.

Q: Can I use functions to represent real-world relationships?

A: Yes, functions can be used to represent real-world relationships, such as the cost of producing a certain product or the temperature of a certain location.

Q: How do I know which operation to perform?

A: The operation to perform depends on the problem you are trying to solve. You need to read the problem carefully and determine which operation is required.

Q: Can I use functions to solve equations?

A: Yes, functions can be used to solve equations by plugging in values and solving for the unknown variable.

Q: How do I graph functions?

A: To graph functions, you can use a graphing calculator or a computer program to visualize the function.

Q: Can I use functions to model real-world phenomena?

A: Yes, functions can be used to model real-world phenomena, such as population growth or disease spread.

Conclusion

References

[1] "Functions" by Khan Academy
[2] "Multiplying Functions" by Mathway
[3] "Adding and Subtracting Functions" by Mathway
[4] "Dividing Functions" by Mathway

FAQs

Q: What is the product of two functions? A: The product of two functions is a new function that is created by multiplying the two functions together.
Q: How do I find the product of two functions? A: To find the product of two functions, you multiply the two functions together, expand the product, simplify the expression, and combine like terms.
Q: What is the difference between the product and the composition of functions? A: The product of two functions is a new function that is created by multiplying the two functions together, while the composition of two functions is a new function that is created by plugging one function into another.

Example Use Cases

Suppose we have a function that represents the cost of producing a certain product, and we want to find the cost of producing a certain quantity of the product. We can use the product of two functions to represent the cost of producing the product.
Suppose we have a function that represents the temperature of a certain location, and we want to find the temperature at a certain time. We can use the composition of two functions to represent the temperature at a certain time.