Select The Correct Answer.Consider These Functions:$\[ \begin{array}{l} F(x) = \frac{1}{3} X^2 + 4 \\ G(x) = 9x - 12 \end{array} \\]What Is The Value Of \($ G(f(x)) $ \)?A. \($ 9x^2 - 24x + 20 $\)B. \[$ 3x

Introduction

In mathematics, composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will explore the composition of functions, specifically the composition of $f(x)$ and $g(x)$, and determine the value of $g(f(x))$.

What are Functions?

A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a rule that assigns to each input exactly one output. In other words, a function takes an input and produces an output.

Composition of Functions

The composition of two functions $f(x)$ and $g(x)$ is denoted by $g(f(x))$. It is a new function that is obtained by applying the function $f(x)$ first and then applying the function $g(x)$ to the result.

Given Functions

We are given two functions:

$f(x) = \frac{1}{3} x^2 + 4$

$g(x) = 9x - 12$

Step 1: Substitute $f(x)$ into $g(x)$

To find the value of $g(f(x))$, we need to substitute $f(x)$ into $g(x)$. This means we will replace $x$ in the function $g(x)$ with the expression $f(x)$.

$g(f(x)) = 9(f(x)) - 12$

Step 2: Simplify the Expression

Now, we need to simplify the expression by substituting the value of $f(x)$ into the equation.

$g(f(x)) = 9(\frac{1}{3} x^2 + 4) - 12$

Step 3: Distribute the 9

Next, we need to distribute the 9 to both terms inside the parentheses.

$g(f(x)) = 9(\frac{1}{3} x^2) + 9(4) - 12$

Step 4: Simplify the Expression

Now, we can simplify the expression by multiplying the 9 with the terms inside the parentheses.

$g(f(x)) = 3x^2 + 36 - 12$

Step 5: Combine Like Terms

Finally, we can combine like terms to simplify the expression.

$g(f(x)) = 3x^2 + 24$

Conclusion

In conclusion, the value of $g(f(x))$ is $3x^2 + 24$. This is the result of composing the functions $f(x)$ and $g(x)$.

Answer

The correct answer is:

A. $3x^2 + 24$

Discussion

The composition of functions is a powerful tool in mathematics that allows us to create new functions by combining existing functions. In this article, we explored the composition of $f(x)$ and $g(x)$ and determined the value of $g(f(x))$. We hope this article has provided a clear understanding of the concept of composition of functions and how to apply it to solve problems.

Example Problems

Here are some example problems that you can try to practice your skills:

1. Find the value of $f(g(x))$ given that $f(x) = 2x + 1$ and $g(x) = x^2 - 3$.
2. Find the value of $g(f(x))$ given that $f(x) = x^2 + 2$ and $g(x) = 3x - 1$.
3. Find the value of $f(g(x))$ given that $f(x) = x^2 - 2$ and $g(x) = 2x + 1$.

Introduction

In our previous article, we explored the concept of composition of functions and determined the value of $g(f(x))$. In this article, we will provide a Q&A guide to help you understand the concept of composition of functions and how to apply it to solve problems.

Q: What is the composition of functions?

A: The composition of two functions $f(x)$ and $g(x)$ is denoted by $g(f(x))$. It is a new function that is obtained by applying the function $f(x)$ first and then applying the function $g(x)$ to the result.

Q: How do I find the value of $g(f(x))$?

A: To find the value of $g(f(x))$, you need to substitute $f(x)$ into $g(x)$. This means you will replace $x$ in the function $g(x)$ with the expression $f(x)$.

Q: What is the order of operations when finding the value of $g(f(x))$?

A: The order of operations when finding the value of $g(f(x))$ is as follows:

1. Substitute $f(x)$ into $g(x)$.
2. Simplify the expression by distributing the terms inside the parentheses.
3. Combine like terms to simplify the expression.

Q: Can I use the composition of functions to solve problems in real-world applications?

A: Yes, the composition of functions can be used to solve problems in real-world applications. For example, in physics, the composition of functions can be used to model the motion of an object under the influence of gravity.

Q: How do I know if a function is a composition of functions?

A: A function is a composition of functions if it can be written as the composition of two or more functions. In other words, if a function can be expressed as $g(f(x))$, then it is a composition of functions.

Q: Can I use the composition of functions to solve problems in mathematics?

A: Yes, the composition of functions can be used to solve problems in mathematics. For example, in calculus, the composition of functions can be used to find the derivative of a function.

Q: What are some common mistakes to avoid when finding the value of $g(f(x))$?

A: Some common mistakes to avoid when finding the value of $g(f(x))$ include:

• Not substituting $f(x)$ into $g(x)$ correctly.
• Not simplifying the expression by distributing the terms inside the parentheses.
• Not combining like terms to simplify the expression.

Q: How can I practice my skills in finding the value of $g(f(x))$?

A: You can practice your skills in finding the value of $g(f(x))$ by trying out example problems. You can also use online resources, such as math websites and apps, to practice your skills.

Conclusion

In conclusion, the composition of functions is a powerful tool in mathematics that allows us to create new functions by combining existing functions. By understanding the concept of composition of functions and how to apply it to solve problems, you can improve your skills in mathematics and apply them to real-world applications.

Example Problems

Here are some example problems that you can try to practice your skills:

1. Find the value of $f(g(x))$ given that $f(x) = 2x + 1$ and $g(x) = x^2 - 3$.
2. Find the value of $g(f(x))$ given that $f(x) = x^2 + 2$ and $g(x) = 3x - 1$.
3. Find the value of $f(g(x))$ given that $f(x) = x^2 - 2$ and $g(x) = 2x + 1$.

We hope these example problems have helped you to understand the concept of composition of functions and how to apply it to solve problems.