Select The Correct Answer.Consider These Functions:${ \begin{array}{l} f(x) = 2x^3 + 3 \ g(x) = X - 4 \end{array} }$What Is The Value Of { G(f(x)) $}$?A. { 2x^3 - 1 $}$ B. { 2x^4 - 4x^3 + 3x - 12 $}$ C.

Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will explore how to evaluate composite functions, specifically the function $g(f(x))$, where $f(x) = 2x^3 + 3$ and $g(x) = x - 4$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Composite Functions

A composite function is a function that is derived from two or more functions. In this case, we have two functions: $f(x) = 2x^3 + 3$ and $g(x) = x - 4$. To evaluate the composite function $g(f(x))$, we need to substitute the expression for $f(x)$ into the function $g(x)$.

Step 1: Substitute the Expression for $f(x)$ into $g(x)$

To evaluate the composite function $g(f(x))$, we need to substitute the expression for $f(x)$ into the function $g(x)$. This means that we will replace the variable $x$ in the function $g(x)$ with the expression $f(x) = 2x^3 + 3$.

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

Step 2: Simplify the Expression

Now that we have substituted the expression for $f(x)$ into the function $g(x)$, we need to simplify the resulting expression. To do this, we will combine like terms and perform any necessary arithmetic operations.

g(f(x)) = 2x^3 - 1

Step 3: Evaluate the Composite Function

Now that we have simplified the expression, we can evaluate the composite function $g(f(x))$. To do this, we will substitute a value for $x$ into the expression and perform any necessary arithmetic operations.

g(f(2)) = 2(2)^3 - 1
g(f(2)) = 2(8) - 1
g(f(2)) = 16 - 1
g(f(2)) = 15

Conclusion

In this article, we have explored how to evaluate composite functions, specifically the function $g(f(x))$, where $f(x) = 2x^3 + 3$ and $g(x) = x - 4$. We have broken down the process into manageable steps and provided a clear explanation of each step. By following these steps, we can evaluate composite functions and gain a deeper understanding of mathematical concepts.

Answer

Based on the steps outlined above, the value of $g(f(x))$ is:

2x^3 - 1

This is the correct answer.

Discussion

The concept of composite functions is a fundamental aspect of mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. By understanding how to evaluate composite functions, we can gain a deeper understanding of mathematical concepts and develop problem-solving skills.

Common Mistakes

When evaluating composite functions, it is common to make mistakes, such as:

• Failing to substitute the expression for $f(x)$ into the function $g(x)$.
• Failing to simplify the resulting expression.
• Failing to evaluate the composite function.

To avoid these mistakes, it is essential to follow the steps outlined above and to take your time when evaluating composite functions.

Real-World Applications

The concept of composite functions has numerous real-world applications, including:

• Physics: Composite functions are used to model complex systems, such as the motion of objects under the influence of gravity.
• Engineering: Composite functions are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Composite functions are used to develop algorithms and data structures, such as sorting and searching algorithms.

By understanding how to evaluate composite functions, we can develop problem-solving skills and gain a deeper understanding of mathematical concepts.

Final Thoughts

Introduction

In our previous article, we explored how to evaluate composite functions, specifically the function $g(f(x))$, where $f(x) = 2x^3 + 3$ and $g(x) = x - 4$. In this article, we will provide a Q&A guide to help you better understand the concept of composite functions and how to evaluate them.

Q: What is a composite function?

A: A composite function is a function that is derived from two or more functions. In this case, we have two functions: $f(x) = 2x^3 + 3$ and $g(x) = x - 4$. To evaluate the composite function $g(f(x))$, we need to substitute the expression for $f(x)$ into the function $g(x)$.

Q: How do I evaluate a composite function?

A: To evaluate a composite function, you need to follow these steps:

1. Substitute the expression for $f(x)$ into the function $g(x)$.
2. Simplify the resulting expression.
3. Evaluate the composite function by substituting a value for $x$ into the expression.

Q: What is the difference between a composite function and a simple function?

A: A composite function is a function that is derived from two or more functions, whereas a simple function is a single function that is not derived from other functions.

Q: Can I use composite functions in real-world applications?

A: Yes, composite functions have numerous real-world applications, including physics, engineering, and computer science. By understanding how to evaluate composite functions, you can develop problem-solving skills and gain a deeper understanding of mathematical concepts.

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

A: Some common mistakes to avoid when evaluating composite functions include:

• Failing to substitute the expression for $f(x)$ into the function $g(x)$.
• Failing to simplify the resulting expression.
• Failing to evaluate the composite function.

Q: How do I know if a function is a composite function?

A: A function is a composite function if it is derived from two or more functions. To determine if a function is a composite function, look for the presence of parentheses or other mathematical operations that indicate the function is derived from other functions.

Q: Can I use composite functions with different types of functions?

A: Yes, composite functions can be used with different types of functions, including polynomial functions, rational functions, and trigonometric functions.

Q: How do I evaluate a composite function with multiple variables?

A: To evaluate a composite function with multiple variables, you need to follow the same steps as before, but with multiple variables. For example, if you have a composite function $g(f(x,y))$, you need to substitute the expression for $f(x,y)$ into the function $g(x,y)$ and then simplify the resulting expression.

Q: Can I use composite functions in calculus?

A: Yes, composite functions are used extensively in calculus, particularly in the study of limits, derivatives, and integrals.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concept of composite functions and how to evaluate them. By following the steps outlined above and avoiding common mistakes, you can develop problem-solving skills and gain a deeper understanding of mathematical concepts.

Commonly Asked Questions

• What is a composite function?
• How do I evaluate a composite function?
• What is the difference between a composite function and a simple function?
• Can I use composite functions in real-world applications?
• What are some common mistakes to avoid when evaluating composite functions?
• How do I know if a function is a composite function?
• Can I use composite functions with different types of functions?
• How do I evaluate a composite function with multiple variables?
• Can I use composite functions in calculus?

Answer Key

• A composite function is a function that is derived from two or more functions.
• To evaluate a composite function, you need to substitute the expression for $f(x)$ into the function $g(x)$, simplify the resulting expression, and then evaluate the composite function.
• A composite function is a function that is derived from two or more functions, whereas a simple function is a single function that is not derived from other functions.
• Yes, composite functions have numerous real-world applications.
• Failing to substitute the expression for $f(x)$ into the function $g(x)$, failing to simplify the resulting expression, and failing to evaluate the composite function.
• A function is a composite function if it is derived from two or more functions.
• Yes, composite functions can be used with different types of functions.
• To evaluate a composite function with multiple variables, you need to substitute the expression for $f(x,y)$ into the function $g(x,y)$ and then simplify the resulting expression.
• Yes, composite functions are used extensively in calculus.